Page i Lectures on Mechanics Jerrold E. Marsden First Published 1992; This Version: October, 2009 Page i Contents Preface v 1 Introduction 1 1.1 The Classical Water Molecule and the Ozone Molecule . . 1 1.2 Lagrangian and Hamiltonian Formulation . . . . . . . . . 3 1.3 The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Geometry, Symmetry and Reduction . . . . . . . . . . . . 12 1.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Geometric Phases . . . . . . . . . . . . . . . . . . . . . . . 19 1.7 The Rotation Group and the Poincare´ Sphere . . . . . . . 26 2 A Crash Course in Geometric Mechanics 29 2.1 Symplectic and Poisson Manifolds . . . . . . . . . . . . . . 29 2.2 The Flow of a Hamiltonian Vector Field . . . . . . . . . . 31 2.3 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . 32 2.4 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . 33 2.5 Lie–Poisson Structures and the Rigid Body . . . . . . . . 34 2.6 The Euler–Poincare´ Equations . . . . . . . . . . . . . . . . 37 2.7 Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . 39 2.8 Symplectic and Poisson Reduction . . . . . . . . . . . . . 42 2.9 Singularities and Symmetry . . . . . . . . . . . . . . . . . 45 2.10 A Particle in a Magnetic Field . . . . . . . . . . . . . . . . 46 3 Tangent and Cotangent Bundle Reduction 49 ii Contents 3.1 Mechanical G-systems . . . . . . . . . . . . . . . . . . . . 50 3.2 The Classical Water Molecule . . . . . . . . . . . . . . . . 52 3.3 The Mechanical Connection . . . . . . . . . . . . . . . . . 56 3.4 The Geometry and Dynamics of Cotangent Bundle Reduc- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.6 Lagrangian Reduction and the Routhian . . . . . . . . . . 72 3.7 The Reduced Euler–Lagrange Equations . . . . . . . . . . 77 3.8 Coupling to a Lie group . . . . . . . . . . . . . . . . . . . 79 4 Relative Equilibria 85 4.1 Relative Equilibria on Symplectic Manifolds . . . . . . . . 85 4.2 Cotangent Relative Equilibria . . . . . . . . . . . . . . . . 88 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4 The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . 95 5 The Energy–Momentum Method 101 5.1 The General Technique . . . . . . . . . . . . . . . . . . . . 101 5.2 Example: The Rigid Body . . . . . . . . . . . . . . . . . . 105 5.3 Block Diagonalization . . . . . . . . . . . . . . . . . . . . 109 5.4 The Normal Form for the Symplectic Structure . . . . . . 114 5.5 Stability of Relative Equilibria for the Double Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6 Geometric Phases 121 6.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . . 121 6.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3 Cotangent Bundle Phases—a Special Case . . . . . . . . . 125 6.4 Cotangent Bundles—General Case . . . . . . . . . . . . . 126 6.5 Rigid Body Phases . . . . . . . . . . . . . . . . . . . . . . 128 6.6 Moving Systems . . . . . . . . . . . . . . . . . . . . . . . . 130 6.7 The Bead on the Rotating Hoop . . . . . . . . . . . . . . 132 7 Stabilization and Control 135 7.1 The Rigid Body with Internal Rotors . . . . . . . . . . . . 135 7.2 The Hamiltonian Structure with Feedback Controls . . . . 136 7.3 Feedback Stabilization of a Rigid Body with a Single Rotor 138 7.4 Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.5 The Kaluza–Klein Description of Charged Particles . . . . 145 7.6 Optimal Control and Yang–Mills Particles . . . . . . . . . 148 8 Discrete Reduction 151 8.1 Fixed Point Sets and Discrete Reduction . . . . . . . . . . 153 8.2 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . 159 8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Contents iii 8.4 Sub-Block Diagonalization with Discrete Symmetry . . . . 165 8.5 Discrete Reduction of Dual Pairs . . . . . . . . . . . . . . 168 9 Mechanical Integrators 173 9.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 173 9.2 Limitations on Mechanical Integrators . . . . . . . . . . . 177 9.3 Symplectic Integrators and Generating Functions . . . . . 179 9.4 Symmetric Symplectic Algorithms Conserve J . . . . . . . 180 9.5 Energy–Momentum Algorithms . . . . . . . . . . . . . . . 182 9.6 The Lie–Poisson Hamilton–Jacobi Equation . . . . . . . . 184 9.7 Example: The Free Rigid Body . . . . . . . . . . . . . . . 187 9.8 PDE Extensions . . . . . . . . . . . . . . . . . . . . . . . 188 10 Hamiltonian Bifurcation 191 10.1 Some Introductory Examples . . . . . . . . . . . . . . . . 191 10.2 The Role of Symmetry . . . . . . . . . . . . . . . . . . . . 198 10.3 The One-to-One Resonance and Dual Pairs . . . . . . . . 203 10.4 Bifurcations in the Double Spherical Pendulum . . . . . . 205 10.5 Continuous Symmetry Groups and Solution Space Singu- larities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.6 The Poincare´–Melnikov Method . . . . . . . . . . . . . . . 207 10.7 The Role of Dissipation . . . . . . . . . . . . . . . . . . . 217 10.8 Double Bracket Dissipation . . . . . . . . . . . . . . . . . 223 References 227 Index 245 Index 246 iv Contents Page v Preface Many of the greatest mathematicians—Euler, Gauss, Lagrange, Rie- mann, Poincare´, Hilbert, Birkhoff, Atiyah, Arnold, Smale—mastered mechanics and many of the greatest advances in mathematics use ideas from mechanics in a fundamental way. Hopefully the revival of it as a basic subject taught to mathematicians will continue and flourish as will the combined geometric, analytic and numerical ap- proach in courses taught in the applied sciences. Anonymous I venture to hope that my lectures may interest engineers, physi- cists, and astronomers as well as mathematicians. If one may accuse mathematicians as a class of ignoring the mathematical problems of the modern physics and astronomy, one may, with no less justice perhaps, accuse physicists and astronomers of ignoring departments of the pure mathematics which have reached a high degree of de- velopment and are fitted to render valuable service to physics and astronomy. It is the great need of the present in mathematical sci- ence that the pure science and those departments of physical science in which it finds its most important applications should again be brought into the intimate association which proved so fruitful in the work of Lagrange and Gauss. Felix Klein, 1896 Mechanics is a broad subject that includes, not only the mechanics of particles and rigid bodies, but also continuum mechanics (fluid mechanics, elasticity, plasma physics, etc), field theory (Maxwell’s equations, the Ein- stein equations etc) and even, as the name implies, quantum mechanics. Rather than compartmentalizing these into separate subjects (and even separate departments in Universities), there is power in seeing these topics as a unified whole. For example, the fact that the flow of a Hamiltonian system consists of canonical transformations becomes a single theorem com- mon to all these topics. Numerical techniques one learns in one branch of mechanics are useful in others. These lectures cover a selection of topics from recent developments in the mathematical approach to mechanics and its applications. In partic- vi Preface ular, we emphasize methods based on symmetry, especially the action of Lie groups, both continuous and discrete, and their associated Noether conserved quantities viewed in the geometric context of momentum maps. In this setting, relative equilibria, the analogue of fixed points for systems without symmetry are especially interesting. In general, relative equilibria are dynamic orbits that are also one-parameter group orbits. For the rota- tion group SO(3), these are uniformly rotating states or, in other words, dynamical motions in steady rotation. Some of the main points treated in the lectures are as follows: • The stability of relative equilibria analyzed using the method of sep- aration of internal and rotational modes, also referred to as the block diagonalization or normal form technique. • Geometric phases, including the phases of Berry and Hannay, are studied using the technique of reduction and reconstruction. • Mechanical integrators, such as numerical schemes that exactly pre- serve the symplectic structure, energy, or the momentum map. • Stabilization & control using methods for mechanical systems. • Bifurcation of relative equilibria in mechanical systems, dealing with the appearance of new relative equilibria and their symmetry break- ing as parameters are varied, and with the development of complex (chaotic) dynamical motions. A unifying theme for many of these aspects is provided by reduction the- ory and the associated mechanical connection for mechanical systems with symmetry. When one does reduction, one sets the corresponding conserved quantity (the momentum map) equal to a constant, and quotients by the subgroup of the symmetry group that leaves this set invariant. One arrives at the reduced symplectic manifold that itself is often a bundle that carries a connection. This connection is induced by a basic ingredient in the the- ory, the mechanical connection on configuration space. This point of view is sometimes called the gauge theory of mechanics. The geometry of reduction and the mechanical connection is an impor- tant ingredient in the decomposition into internal and rotational modes in the block diagonalization method, a powerful method for analyzing the stability and bifurcation of relative equilibria. The holonomy of the con- nection on the reduction bundle gives geometric phases. When stability of a relative equilibrium is lost, one can get bifurcation, solution symmetry breaking , instability and chaos. The notion of system symmetry break- ing (also called forced symmetry breaking) in which not only the solutions, but the equations themselves lose symmetry, is also important but here is treated only by means of some simple examples. Preface vii Two related topics that are discussed are control and mechanical inte- grators. One would like to be able to control the geometric phases with the aim of, for example, controlling the attitude of a rigid body with internal rotors. With mechanical integrators one is interested in designing numeri- cal integrators that exactly preserve the conserved momentum (say angular momentum) and either the energy or symplectic structure, for the purpose of accurate long time integration of mechanical systems. Such integrators are becoming popular methods as their performance gets tested in specific applications. We include a chapter on this topic that is meant to be a basic introduction to the theory, but not the practice of these algorithms. This work proceeds at a reasonably advanced level but has the corre- sponding advantage of a shorter length. The exposition is intended to be, as the title suggests, in lecture style and is not meant to be a complete treatise. For a more detailed exposition of many of these topics suitable for beginning students in the subject, see Marsden and Ratiu [1999]. There are also numerous references to the literature where the reader can find additional information. The work of many of my colleagues from around the world is drawn upon in these lectures and is hereby gratefully acknowledged. In this regard, I es- pecially thank Mark Alber, Vladimir Arnold, Judy Arms, John Ball, Tony Bloch, David Chillingworth, Richard Cushman, Michael Dellnitz, Arthur Fischer, Mark Gotay, Marty Golubitsky, John Harnad, Aaron Hershman, Darryl Holm, Phil Holmes, John Guckenheimer, Jacques Hurtubise, Sameer Jalnapurkar, Vivien Kirk, Wang-Sang Koon, P.S. Krishnaprasad, Debbie Lewis, Robert Littlejohn, Ian Melbourne, Vincent Moncrief, Richard Mont- gomery, George Patrick, Tom Posbergh, Tudor Ratiu, Alexi Reyman, Glo- ria Sanchez de Alvarez, Shankar Sastry, Ju¨rgen Scheurle, Mary Silber, Juan Simo, Ian Stewart, Greg Walsh, Steve Wan, Alan Weinstein, Shmuel Weiss- man, Steve Wiggins, and Brett Zombro. The work of others is cited at appropriate points in the text. I would like to especially thank David Chillingworth for organizing the LMS lecture series in Southampton, April 15–19, 1991 that acted as a ma- jor stimulus for preparing the written version of these notes. I would like to also thank the Mathematical Sciences Research Institute and especially Alan Weinstein and Tudor Ratiu at Berkeley for arranging a preliminary set of lectures along these lines in April, 1989, and Francis Clarke at the Centre de Recherches Mathe´matique in Montre´al for his hospitality during the Aisenstadt lectures in the fall of 1989. Thanks are also due to Phil Holmes and John Guckenheimer at Cornell, the Mathematical Sciences Institute, and to David Sattinger and Peter Olver at the University of Minnesota, and the Institute for Mathematics and its Applications, where several of these talks were given in various forms. I also thank the Hum- boldt Stiftung of Germany, Ju¨rgen Scheurle and Klaus Kirchga¨ssner who provided the opportunity and resources needed to put the lectures to paper during a pleasant and fruitful stay in Hamburg and Blankenese during the viii Preface first half of 1991. I also acknowledge a variety of research support from NSF and DOE that helped make the work possible. I thank several partic- ipants of the lecture series and other colleagues for their useful comments and corrections. I especially thank Hans Peter Kruse, Oliver O’Reilly, Rick Wicklin, Brett Zombro and Florence Lin in this respect. Very special thanks go to Barbara for typesetting the lectures and for her support in so many ways. Our late cat Thomas also deserves thanks for his help with our understanding of 180◦ cat maneuvers. This work was not responsible for his unfortunate fall from the roof (resulting in a broken paw), but his feat did prove that cats can execute 90◦ attitude control as well. Jerry Marsden Pasadena, California Summer, 2004 0 Preface Page 1 1 Introduction This chapter gives an overview of some of the topics that will be covered so the reader can get a coherent picture of the types of problems and associated mathematical structures that will be developed.1 1.1 The Classical Water Molecule and the Ozone Molecule An example that will be used to illustrate various concepts throughout these lectures is the classical (non-quantum) rotating “water molecule”. This system, shown in Figure 1.1.1, consists of three particles interacting by interparticle conservative forces (think of springs connecting the par- ticles, for example). The total energy of the system, which will be taken as our Hamiltonian, is the sum of the kinetic and potential energies, while the Lagrangian is the difference of the kinetic and potential energies. The interesting special case of three equal masses gives the “ozone” molecule. We use the term “water molecule” mainly for terminological convenience. The full problem is of course the classical three body problem in space. However, thinking of it as a rotating system evokes certain constructions that we wish to illustrate. 1We are grateful to Oliver O’Reilly, Rick Wicklin, and Brett Zombro for providing a helpful draft of the notes for an early version of this lecture. 2 1. Introduction z y x m M m Figure 1.1.1. The rotating and vibrating water molecule. Imagine this mechanical system rotating in space and, simultaneously, undergoing vibratory, or internal motions. We can ask a number of ques- tions: • How does one set up the equations of motion for this system? • Is there a convenient way to describe steady rotations? Which of these are stable? When do bifurcations occur? • Is there a way to separate the rotational from the internal motions? • How do vibrations affect overall rotations? Can one use them to con- trol overall rotations? To stabilize otherwise unstable motions? • Can one separate symmetric (the two hydrogen atoms moving as mir- ror images) and non-symmetric vibrations using a discrete symmetry? • Does a deeper understanding of the classical mechanics of the water molecule help with the corresponding quantum problem? It is interesting that despite the old age of classical mechanics, new and deep insights are coming to light by combining the rich heritage of knowl- edge already well founded by masters like Newton, Euler, Lagrange, Jacobi, Laplace, Riemann and Poincare´, with the newer techniques of geometry and qualitative analysis of people like Arnold and Smale. I hope that already the classical water molecule and related systems will convey some of the spirit of modern research in geometric mechanics. The water molecule is in fact too hard an example to carry out in as much detail as one would like, although it illustrates some of the general theory quite nicely. A simpler example for which one can get more detailed 1.2 Lagrangian and Hamiltonian Formulation 3 information (about relative equilibria and their bifurcations, for example) is the double spherical pendulum . Here, instead of the symmetry group being the full (non-Abelian) rotation group SO(3), it is the (Abelian) group S1 of rotations about the axis of gravity. The double pendulum will also be used as a thread through the lectures. The results for this example are drawn from Marsden and Scheurle [1993a, 1993b]. To make similar progress with the water molecule, one would have to deal with the already complex issue of finding a reasonable model for the interatomic potential. There is a large literature on this going back to Darling and Dennison [1940] and Sorbie and Murrell [1975]. For some of the recent work that might be important for the present approach, and for more references, see Xiao and Kellman [1989] and Li, Xiao, and Kellman [1990]. Other useful references are Haller [1999], Montaldi and Roberts [1999] and Tanimura and Iwai [1999]. The special case of the ozone molecule with its three equal masses is also of great interest, not only for environmental reasons, but because this molecule has more symmetry than the water molecule. In fact, what we learn about the water molecule can be used to study the ozone molecule by putting m = M . A big change that has very interesting consequences is the fact that the discrete symmetry group is enlarged from “reflections” Z2 to the “symmetry group of a triangle” D3. This situation is also of interest in chemistry for things like molecular control by using laser beams to control the potential in which the molecule finds itself. Some believe that, together with ideas from semiclassical quantum mechanics, the study of this system as a classical system provides useful information. We refer to Pierce, Dahleh, and Rabitz [1988], Tannor [1989] and Tannor and Jin [1991] for more information and literature leads. 1.2 Lagrangian and Hamiltonian Formulation Lagrangian Formalism. Around 1790, Lagrange introduced general- ized coordinates (q1, . . . , qn) and their velocities (q˙q, . . . , q˙n) to describe the state of a mechanical system. Motivated by covariance (coordinate inde- pendence) considerations, he introduced the Lagrangian L(qi, q˙i), which is often the kinetic energy minus the potential energy, and proposed the equations of motion in the form d dt ∂L ∂q˙i − ∂L ∂qi = 0, (1.2.1) 4 1. Introduction called the Euler–Lagrange equations. About 1830, Hamilton realized how to obtain these equations from a variational principle δ ∫ b a L(qi(t), q˙i(t))dt = 0, (1.2.2) called the principle of critical action , in which the variation is over all curves with two fixed endpoints and with a fixed time interval [a, b]. Curiously, Lagrange knew the more sophisticated principle of least action (that has the constraint of conservation of energy added), but not the proof of the equivalence of (1.2.1) and (1.2.2), which is simple if we assume the appropriate regularity and is as follows. Let q(t, ) be a family of curves with q(t) = q(t, 0) and let the variation be defined by δq(t) = d d q(t, ) ∣∣∣∣ =0 . (1.2.3) Note that, by equality of mixed partial derivatives, δq˙(t) = δ˙q(t). Differentiating ∫ b a L(qi(t, ), q˙i(t, ))dt in at = 0 and using the chain rule gives δ ∫ b a Ldt = ∫ b a ( ∂L ∂qi δqi + ∂L ∂q˙i δq˙i ) dt = ∫ b a ( ∂L ∂qi δqi − d dt ∂L ∂q˙i δqi ) dt where we have integrated the second term by parts and have used δqi(a) = δqi(b) = 0. Since δqi(t) is arbitrary except for the boundary conditions, the equivalence of (1.2.1) and (1.2.2) becomes evident. The collection of pairs (q, q˙) may be thought of as elements of the tangent bundle TQ of configuration space Q. We also call TQ the velocity phase space . One of the great achievements of Lagrange was to realize that (1.2.1) and (1.2.2) make intrinsic (coordinate independent) sense; today we would say that Lagrangian mechanics can be formulated on manifolds. For me- chanical systems like the rigid body, coupled structures etc., it is essential that Q be taken to be a manifold and not just Euclidean space. The Hamiltonian Formalism. If we perform the Legendre trans- form, that is, change variables to the cotangent bundle T ∗Q by pi = ∂L ∂q˙i (assuming this is an invertible change of variables) and let the Hamilto- nian be defined by H(qi, pi) = piq˙i − L(qi, q˙i) (1.2.4) 1.3 The Rigid Body 5 (summation on repeated indices understood), then the Euler–Lagrange equations become Hamilton’s equations q˙i = ∂H ∂pi , p˙i = −∂H ∂qi , i = 1, . . . , n. (1.2.5) The symmetry in these equations leads to a rich geometric structure. 1.3 The Rigid Body As we just saw, the equations of motion for a classical mechanical system with n degrees of freedom may be written as a set of first order equations in Hamiltonian form: q˙i = ∂H ∂pi , p˙i = −∂H ∂qi , i = 1, . . . , n. The configuration coordinates (q1, . . . , qn) and momenta (p1, . . . , pn) to- gether define the system’s instantaneous state, which may also be regarded as the coordinates of a point in the cotangent bundle T ∗Q, the systems (momentum) phase space . The Hamiltonian function H(q, p) defines the system and, in the absence of constraining forces and time dependence, is the total energy of the system. The phase space for the water molecule is R18 (perhaps with collision points removed) and the Hamiltonian is the kinetic plus potential energies. Recall that the set of all possible spatial positions of bodies in the sys- tem is their configuration space Q. For example, the configuration space for the water molecule is R9 and for a three dimensional rigid body mov- ing freely in space is SE(3), the six dimensional group of Euclidean (rigid) transformations of three-space, that is, all possible rotations and transla- tions. If translations are ignored and only rotations are considered, then the configuration space is SO(3). As another example, if two rigid bodies are connected at a point by an idealized ball-in-socket joint, then to specify the position of the bodies, we must specify a single translation (since the bodies are coupled) but we need to specify two rotations (since the two bod- ies are free to rotate in any manner). The configuration space is therefore SE(3)× SO(3). This is already a fairly complicated object, but remember that one must keep track of both positions and momenta of each compo- nent body to formulate the system’s dynamics completely. If Q denotes the configuration space (only positions), then the corresponding phase space P (positions and momenta) is the manifold known as the cotangent bundle of Q, which is denoted by T ∗Q. One of the important ways in which the modern theory of Hamiltonian systems generalizes the classical theory is by relaxing the requirement of using canonical phase space coordinate systems, that is, coordinate systems 6 1. Introduction in which the equations of motion have the form (1.2.5) above. Rigid body dynamics, celestial mechanics, fluid and plasma dynamics, nonlinear elas- todynamics and robotics provide a rich supply of examples of systems for which canonical coordinates can be unwieldy and awkward. The free mo- tion of a rigid body in space was treated by Euler in the eighteenth century and yet it remains remarkably rich as an illustrative example. Notice that if our water molecule has stiff springs between the atoms, then it behaves nearly like a rigid body. One of our aims is to bring out this behavior. The rigid body problem in its primitive formulation has the six di- mensional configuration space SE(3). This means that the phase space, T ∗ SE(3) is twelve dimensional. Assuming that no external forces act on the body, conservation of linear momentum allows us to solve for the com- ponents of the position and momentum vectors of the center of mass. Re- duction to the center of mass frame, which we will work out in detail for the classical water molecule, reduces one to the case where the center of mass is fixed, so only SO(3) remains. Each possible orientation corresponds to an element of the rotation group SO(3) which we may therefore view as a configuration space for all “non-trivial” motions of the body. Euler for- mulated a description of the body’s orientation in space in terms of three angles between axes that are either fixed in space or are attached to sym- metry planes of the body’s motion. The three Euler angles, ψ,ϕ and θ are generalized coordinates for the problem and form a coordinate chart for SO(3). However, it is simpler and more convenient to proceed intrinsically as follows. We regard the element A ∈ SO(3) giving the configuration of the body as a map of a reference configuration B ⊂ R3 to the current configuration A(B). The map A takes a reference or label point X ∈ B to a current point x = A(X) ∈ A(B). For a rigid body in motion, the matrix A becomes time dependent and the velocity of a point of the body is x˙ = A˙X = A˙A−1x. Since A is an orthogonal matrix, we can write x˙ = A˙A−1x = ω × x, (1.3.1) which defines the spatial angular velocity vector ω. The corresponding body angular velocity is defined by Ω = A−1ω, (1.3.2) so that Ω is the angular velocity as seen in a body fixed frame. The kinetic energy is the usual expression K = 1 2 ∫ B ρ(X)‖A˙X‖2 d3X, (1.3.3) where ρ is the mass density. Since ‖A˙X‖ = ‖ω × x‖ = ‖A−1(ω × x)‖ = ‖Ω×X‖, 1.3 The Rigid Body 7 the kinetic energy is a quadratic function of Ω. Writing K = 1 2 ΩT IΩ (1.3.4) defines the (time independent) moment of inertia tensor I, which, if the body does not degenerate to a line, is a positive definite 3× 3 matrix, or better, a quadratic form. Its eigenvalues are called the principal mo- ments of inertia. This quadratic form can be diagonalized, and provided the eigenvalues are distinct, uniquely defines the principal axes. In this basis, we write I = diag(I1, I2, I3). Every calculus text teaches one how to compute moments of inertia! From the Lagrangian point of view, the precise relation between the motion in A space and in Ω space is as follows. 1.3.1 Theorem. The curve A(t) ∈ SO(3) satisfies the Euler–Lagrange equations for L(A, A˙) = 1 2 ∫ B ρ(X)‖A˙X‖2d3X (1.3.5) if and only if Ω(t) defined by A−1A˙v = Ω × v for all v ∈ R3 satisfies Euler’s equations: IΩ˙ = IΩ× Ω. (1.3.6) Moreover, this equation is equivalent to conservation of the spatial angular momentum: d dt pi = 0 (1.3.7) where pi = AIΩ. Probably the simplest way to prove this is to use variational principles. We already saw that A(t) satisfies the Euler–Lagrange equations if and only if δ ∫ Ldt = 0. Let l(Ω) = 12 (IΩ) · Ω so that l(Ω) = L(A, A˙) if A and Ω are related as above. To see how we should transform the variational principle for L, we differentiate the relation A−1A˙v = Ω× v (1.3.8) with respect to a parameter describing a variation ofA, as we did in (1.2.3), to get −A−1δAA−1A˙v +A−1δA˙v = δΩ× v. (1.3.9) Let the skew matrix Σˆ be defined by Σˆ = A−1δA (1.3.10) and define the associated vector Σ by Σˆv = Σ× v. (1.3.11) 8 1. Introduction From (1.3.10) we get ˙ˆΣ = −A−1A˙A−1δA+A−1δA˙, and so A−1δA˙ = ˙ˆΣ +A−1A˙Σˆ. (1.3.12) Substituting (1.3.8), (1.3.10), and (1.3.12) into (1.3.9) gives −ΣˆΩˆv + ˙ˆΣv + ΩˆΣˆv = δ̂Ωv; that is, δ̂Ω = ˙ˆΣ + [Ωˆ, Σˆ]. (1.3.13) Now one checks the identity [Ωˆ, Σˆ] = (Ω× Σ)ˆ (1.3.14) by using Jacobi’s identity for the cross product. Thus, (1.3.13) gives δΩ = Σ˙ + Ω× Σ. (1.3.15) These calculations prove the following 1.3.2 Theorem. The variational principle δ ∫ b a Ldt = 0 (1.3.16) on SO(3) is equivalent to the reduced variational principle δ ∫ b a l dt = 0 (1.3.17) on R3 where the variations δΩ are of the form (1.3.15) with Σ(a) = Σ(b) = 0. To complete the proof of Theorem 1.3.1, it suffices to work out the equations equivalent to the reduced variational principle (1.3.17). Since l(Ω) = 12 〈IΩ,Ω〉, and I is symmetric, we get δ ∫ b a l dt = ∫ b a 〈IΩ, δΩ〉dt = ∫ b a 〈IΩ, Σ˙ + Ω× Σ〉dt = ∫ b a [〈 − d dt IΩ,Σ 〉 + 〈IΩ,Ω× Σ〉 ] dt = ∫ b a 〈 − d dt IΩ + IΩ× Ω,Σ 〉 dt 1.3 The Rigid Body 9 where we have integrated by parts and used the boundary conditions Σ(b) = Σ(a) = 0. Since Σ is otherwise arbitrary, (1.3.17) is equivalent to − d dt (IΩ) + IΩ× Ω = 0, which are Euler’s equations. As we shall see in Chapter 2, this calculation is a special case of a pro- cedure valid for any Lie group and, as such, leads to the Euler–Poincare´ equations; (Poincare´ [1901]). The body angular momentum is defined, analogous to linear momen- tum p = mv, as Π = IΩ so that in principal axes, Π = (Π1,Π2,Π3) = (I1Ω1, I2Ω2, I3Ω3). As we have seen, the equations of motion for the rigid body are the Euler– Lagrange equations for the Lagrangian L equal to the kinetic energy, but regarded as a function on T SO(3) or equivalently, Hamilton’s equations with the Hamiltonian equal to the kinetic energy, but regarded as a function on the cotangent bundle of SO(3). In terms of the Euler angles and their conjugate momenta, these are the canonical Hamilton equations, but as such they are a rather complicated set of six ordinary differential equations. Assuming that no external moments act on the body, the spatial angular momentum vector pi = AΠ is conserved in time. As we shall recall in Chapter 2, this follows by general considerations of symmetry, but it can also be checked directly from Euler’s equations: dpi dt = A˙IΩ +A(IΩ× Ω) = A(A−1A˙IΩ + IΩ× Ω) = A(Ω× IΩ + IΩ× Ω) = 0. Thus, pi is constant in time. In terms of Π, the Euler equations read Π˙ = Π× Ω, or, equivalently Π˙1 = I2 − I3 I2I3 Π2Π3, Π˙2 = I3 − I1 I3I1 Π3Π1, Π˙3 = I1 − I2 I1I2 Π1Π2. (1.3.18) Arnold [1966] clarified the relationships between the various representations (body, space, Euler angles) of the equations and showed how the same ideas apply to fluid mechanics as well. 10 1. Introduction Viewing (Π1,Π2,Π3) as coordinates in a three dimensional vector space, the Euler equations are evolution equations for a point in this space. An integral (constant of motion) for the system is given by the magnitude of the total angular momentum vector: ‖Π‖2 = Π21+Π21+Π21. This follows from conservation of pi and the fact that ‖pi‖ = ‖Π‖ or can be verified directly from the Euler equations by computing the time derivative of ‖Π‖2 and observing that the sum of the coefficients in (1.3.18) is zero. Because of conservation of ‖Π‖, the evolution in time of any initial point Π(0) is constrained to the sphere ‖Π‖2 = ‖Π(0)‖2 = constant. Thus we may view the Euler equations as describing a two dimensional dynamical system on an invariant sphere. This sphere is the reduced phase space for the rigid body equations. In fact, this defines a two dimensional system as a Hamiltonian dynamical system on the two-sphere S2. The Hamiltonian structure is not obvious from Euler’s equations because the description in terms of the body angular momentum is inherently non-canonical. As we shall see in §1.4 and in more detail in Chapter 4, the theory of Hamiltonian systems may be generalized to include Euler’s formulation. The Hamilto- nian for the reduced system is H = 1 2 〈Π, I−1Π〉 = 1 2 ( Π21 I1 + Π22 I2 + Π23 I3 ) (1.3.19) and we shall show how this function allows us to recover Euler’s equa- tions (1.3.18). Since solutions curves are confined to the level sets of H (which are in general ellipsoids) as well as to the invariant spheres ‖Π‖ = constant, the intersection of these surfaces are precisely the trajectories of the rigid body, as shown in Figure 1.3.1. On the reduced phase space, dynamical fixed points are called relative equilibria . These equilibria correspond to periodic orbits in the unreduced phase space, specifically to steady rotations about a principal inertial axis. The locations and stability types of the relative equilibria for the rigid body are clear from Figure 1.3.1. The four points located at the intersections of the invariant sphere with the Π1 and Π2 axes correspond to pure rotational motions of the body about its major and minor principal axes. These mo- tions are stable, whereas the other two relative equilibria corresponding to rotations about the intermediate principal axis are unstable. In Chapters 4 and 5 we shall see how the stability analysis for a large class of more complicated systems can be simplified through a careful choice of non-canonical coordinates. We managed to visualize the trajectories of the rigid body without doing any calculations, but this is because the rigid body is an especially simple system. Problems like the rotating water molecule will prove to be more challenging. Not only is the rigid body problem in- tegrable (one can write down the solution in terms of integrals), but the problem reduces in some sense to a two dimensional manifold and allows questions about trajectories to be phrased in terms of level sets of integrals. Many Hamiltonian systems are not integrable and trajectories are chaotic 1.3 The Rigid Body 11 Figure 1.3.1. Phase portrait for the rigid body. The magnitude of the angular momentum vector determines a sphere. The intersection of the sphere with the ellipsoids of constant Hamiltonian gives the trajectories of the rigid body. and are often studied numerically. The fact that we were able to reduce the number of dimensions in the problem (from twelve to two) and the fact that this reduction was accomplished by appealing to the non-canonical coordi- nates Ω or Π turns out to be a general feature for Hamiltonian systems with symmetry. The reduction procedure may be applied to non-integrable or chaotic systems, just as well as to integrable ones. In a Hamiltonian context, non-integrability is generally taken to mean that any analytic constant of motion is a function of the Hamiltonian. We will not attempt to formulate a general definition of chaos, but rather use the term in a loose way to refer to systems whose motion is so complicated that long-term prediction of dynamics is impossible. It can sometimes be very difficult to establish whether a given system is chaotic or non-integrable. Sometimes theoret- ical tools such as “Melnikov’s method” (see Guckenheimer and Holmes [1983] and Wiggins [1988]) are available. Other times, one resorts to nu- merics or direct observation. For instance, numerical integration suggests that irregular natural satellites such as Saturn’s moon, Hyperion, tumble in their orbits in a highly irregular manner (see Wisdom, Peale, and Mignard [1984]). The equations of motion for an irregular body in the presence of a non-uniform gravitational field are similar to the Euler equations except that there is a configuration-dependent gravitational moment term in the equations that presumably render the system non-integrable. The evidence that Hyperion tumbles chaotically in space leads to diffi- culties in numerically modelling this system. The manifold SO(3) cannot be covered by a single three dimensional coordinate chart such as the Eu- ler angle chart (see §1.7). Hence an integration algorithm using canonical 12 1. Introduction variables must employ more than one coordinate system, alternating be- tween coordinates on the basis of the body’s current configuration. For a body that tumbles in a complicated fashion, the body’s configuration might switch from one chart of SO(3) to another in a short time interval, and the computational cost for such a procedure could be prohibitive for long time integrations. This situation is worse still for bodies with internal degrees of freedom like our water molecule, robots, and large-scale space structures. Such examples point out the need to go beyond canonical formulations. 1.4 Geometry, Symmetry and Reduction We have emphasized the distinction between canonical and non-canonical coordinates by contrasting Hamilton’s (canonical) equations with Euler’s equations. We may view this distinction from a different perspective by introducing Poisson bracket notation. Given two smooth (C∞) real-valued functions F and K defined on the phase space of a Hamiltonian system, define the canonical Poisson bracket of F and K by {F,K} = n∑ i=1 ( ∂F ∂qi ∂K ∂pi − ∂K ∂qi ∂F ∂pi ) (1.4.1) where (qi, pi) are conjugate pairs of canonical coordinates. If H is the Hamiltonian function for the system, then the formula for the Poisson bracket yields the directional derivative of F along the flow of Hamilton’s equations; that is, F˙ = {F,H}. (1.4.2) In particular, Hamilton’s equations are recovered if we let F be each of the canonical coordinates in turn: q˙i = {qi, H} = ∂H ∂pi , p˙i = {pi, H} = −∂H ∂qi . Once H is specified, the chain rule shows that the statement “F˙ = {F,H} for all smooth functions F” is equivalent to Hamilton’s equations. In fact, it tells how any function F evolves along the flow. This representation of the canonical equations of motion suggests a gen- eralization of the bracket notation to cover non-canonical formulations. As an example, consider Euler’s equations. Define the following non-canonical rigid body bracket of two smooth functions F and K on the angular momentum space: {F,K} = −Π · (∇F ×∇K), (1.4.3) where {F,K} and the gradients of F and K are evaluated at the point Π = (Π1,Π2,Π3). The notation in (1.4.3) is that of the standard scalar 1.4 Geometry, Symmetry and Reduction 13 triple product operation in R3. If H is the rigid body Hamiltonian (see (1.3.18)) and F is, in turn, allowed to be each of the three coordinate functions Πi, then the formula F˙ = {F,H} yields the three Euler equations. The non-canonical bracket corresponding to the reduced free rigid body problem is an example of what is known as a Lie–Poisson bracket. In Chapter 2 we shall see how to generalize this to any Lie algebra. Other bracket operations have been developed to handle a wide variety of Hamil- tonian problems in non-canonical form, including some problems outside of the framework of traditional Newtonian mechanics (see for instance, Arnold [1966], Marsden, Weinstein, Ratiu, Schmid, and Spencer [1982] and Holm, Marsden, Ratiu, and Weinstein [1985]). In Hamiltonian dynamics, it is essential to distinguish features of the dynamics that depend on the Hamiltonian function from those that depend only on properties of the phase space. The generalized bracket operation is a geometric invariant in the sense that it depends only on the structure of the phase space. The phase spaces arising in mechanics often have an additional geometric structure closely related to the Poisson bracket. Specifically, they may be equipped with a special differential two-form called the symplectic form . The symplectic form defines the geometry of a symplectic manifold much as the metric tensor defines the geometry of a Riemannian manifold. Bracket operations can be defined entirely in terms of the symplectic form without reference to a particular coordinate system. The classical concept of a canonical transformation can also be given a more geometric definition within this framework. A canonical transfor- mation is classically defined as a transformation of phase space that takes one canonical coordinate system to another. The invariant version of this concept is a symplectic map, a smooth map of a symplectic manifold to itself that preserves the symplectic form or, equivalently, the Poisson bracket operation. The geometry of symplectic manifolds is an essential ingredient in the formulation of the reduction procedure for Hamiltonian systems with sym- metry. We now outline some important ingredients of this procedure and will go into this in more detail in Chapters 2 and 3. In Euler’s problem of the free rotation of a rigid body in space (assuming that we have already ex- ploited conservation of linear momentum), the six dimensional phase space is T ∗ SO(3)—the cotangent bundle of the three dimensional rotation group. This phase space T ∗ SO(3) is often parametrized by three Euler angles and their conjugate momenta. The reduction from six to two dimensions is a consequence of two essential features of the problem: 1. Rotational invariance of the Hamiltonian, and 2. The existence of a corresponding conserved quantity, the spatial an- gular momentum. 14 1. Introduction These two conditions are generalized to arbitrary mechanical systems with symmetry in the general reduction theory of Marsden and Weinstein [1974] (see also Meyer [1973]), which was inspired by the seminal works of Arnold [1966] and Smale [1970]. In this theory, one begins with a given phase space that we denote P . We assume there is a group G of symmetry trans- formations of P that transform P to itself by canonical transformation. Generalizing 2, we use the symmetry group to generate a vector-valued conserved quantity denoted J and called the momentum map. Analogous to the set where the total angular momentum has a given value, we consider the set of all phase space points where J has a given value µ; that is, the µ-level set for J. The analogue of the two dimensional body angular momentum sphere in Figure 1.3.1 is the reduced phase space , denoted Pµ that is constructed as follows: Pµ is the µ-level set for J on which any two points that can be transformed one to the other by a group transformation are identified. The reduction theorem states that Pµ inherits the symplectic (and hence Poisson bracket) structure from that of P , so it can be used as a new phase space. Also, dynamical trajectories of the Hamiltonian H on P determine corresponding trajectories on the reduced space. This new dynamical system is, naturally, called the reduced system . The trajectories on the sphere in Figure 1.3.1 are the reduced trajectories for the rigid body problem. We saw that steady rotations of the rigid body correspond to fixed points on the reduced manifold, that is, on the body angular momentum sphere. In general, fixed points of the reduced dynamics on Pµ are called relative equilibria , following terminology introduced by Poincare´ [1885]. The re- duction process can be applied to the system that models the motion of the moon Hyperion, to spinning tops, to fluid and plasma systems, and to systems of coupled rigid bodies. For example, if our water molecule is undergoing steady rotation, with the internal parts not moving relative to each other, this will be a relative equilibrium of the system. An oblate Earth in steady rotation is a relative equilibrium for a fluid-elastic body. In general, the bigger the symmetry group, the richer the supply of relative equilibria. Fluid and plasma dynamics represent one of the interesting areas to which these ideas apply. In fact, already in the original paper of Arnold [1966], fluids are studied using methods of geometry and reduction. In particular, it was this method that led to the first analytical nonlinear stability result for ideal flow, namely the nonlinear version of the Rayleigh inflection point criterion in Arnold [1969]. These ideas were continued in 1.5 Stability 15 Ebin and Marsden [1970] with the major result that the Euler equations in material representation are governed by a smooth vector field in the Sobolev Hs topology, with applications to convergence results for the zero viscosity limit. In Morrison [1980] and Marsden and Weinstein [1982] the Hamiltonian structure of the Maxwell–Vlasov equations of plasma physics was found and in Holm, Kupershmidt, and Levermore [1985] the stability for these equations along with other fluid and plasma applications was investigated. In fact, the literature on these topics is now quite extensive, and we will not attempt a survey here. We refer to Marsden and Ratiu [1999] for more details. However, some of the basic techniques behind these applications are discussed in the sections that follow. 1.5 Stability There is a standard procedure for determining the stability of equilibria of an ordinary differential equation x˙ = f(x) (1.5.1) where x = (x1, . . . , xn) and f is smooth. Equilibria are points xe such that f(xe) = 0; that is, points that are fixed in time under the dynamics. By stability of the fixed point xe we mean that any solution to x˙ = f(x) that starts near xe remains close to xe for all future time. A traditional method of ascertaining the stability of xe is to examine the first variation equation ξ˙ = df(xe)ξ (1.5.2) where df(xe) is the Jacobian of f at xe, defined to be the matrix of partial derivatives df(xe) = [ ∂f i ∂xj ] x=xe . (1.5.3) Liapunov’s theorem If all the eigenvalues of df(xe) lie in the strict left half plane, then the fixed point xe is stable. If any of the eigenvalues lie in the right half plane, then the fixed point is unstable. For Hamiltonian systems, the eigenvalues come in quartets that are sym- metric about the origin, and so they cannot all lie in the strict left half plane. (See, for example, Marsden and Ratiu [1999] for the proof of this assertion.) Thus, the above form of Liapunov’s theorem is not appropriate to deduce whether or not a fixed point of a Hamiltonian system is stable. When the Hamiltonian is in canonical form, one can use a stability test for fixed points due to Lagrange and Dirichlet. This method starts with the observation that for a fixed point (qe, pe) of such a system, ∂H ∂q (qe, pe) = ∂H ∂p (qe, pe) = 0. 16 1. Introduction Hence the fixed point occurs at a critical point of the Hamiltonian. Lagrange-Dirichlet Criterion If the 2n × 2n matrix δ2H of second partial derivatives, (the second variation) is either positive or negative definite at (qe, pe) then it is a stable fixed point. The proof is very simple. Consider the positive definite case. Since H has a nondegenerate minimum at ze = (qe, pe), Taylor’s theorem with remainder shows that its level sets near ze are bounded inside and outside by spheres of arbitrarily small radius. Since energy is conserved, solutions stay on level surfaces of H, so a solution starting near the minimum has to stay near the minimum. For a Hamiltonian of the form kinetic plus potential V , critical points occur when pe = 0 and qe is a critical point of the potential of V . The Lagrange-Dirichlet Criterion then reduces to asking for a non-degenerate minimum of V . In fact, this criterion was used in one of the classical problems of the 19th century: the problem of rotating gravitating fluid masses. This problem was studied by Newton, MacLaurin, Jacobi, Riemann, Poincare´ and others. The motivation for its study was in the conjectured birth of two planets by the splitting of a large mass of solidifying rotating fluid. Riemann [1860], Routh [1877] and Poincare´ [1885, 1892, 1901] were major contributors to the study of this type of phenomenon and used the potential energy and angular momentum to deduce the stability and bifurcation. The Lagrange-Dirichlet method was adapted by Arnold [1966, 1969] into what has become known as the energy–Casimir method . Arnold ana- lyzed the stability of stationary flows of perfect fluids and arrived at an explicit stability criterion when the configuration space Q for the Hamilto- nian of this system is the symmetry group G of the mechanical system. A Casimir function C is one that Poisson commutes with any function F defined on the phase space of the Hamiltonian system, that is, {C,F} = 0. (1.5.4) Large classes of Casimirs can occur when the reduction procedure is per- formed, resulting in systems with non-canonical Poisson brackets. For ex- ample, in the case of the rigid body discussed previously, if Φ is a function of one variable and µ is the angular momentum vector in the inertial coor- dinate system, then C(µ) = Φ(‖µ‖2) (1.5.5) is readily checked to be a Casimir for the rigid body bracket (1.3.3). Energy–Casimir method Choose C such that H + C has a critical point at an equilibrium ze and compute the second variation δ2(H + C)(ze). If this matrix is positive or negative definite, then the equilibrium ze is stable. 1.5 Stability 17 When the phase space is obtained by reduction, the equilibrium ze is called a relative equilibrium of the original Hamiltonian system. The energy–Casimir method has been applied to a variety of problems including problems in fluids and plasmas (Holm, Kupershmidt, and Lev- ermore [1985]) and rigid bodies with flexible attachments (Krishnaprasad and Marsden [1987]). If applicable, the energy–Casimir method may per- mit an explicit determination of the stability of the relative equilibria. It is important to remember, however, that these techniques give stability in- formation only. As such one cannot use them to infer instability without further investigation. The energy–Casimir method is restricted to certain types of systems, since its implementation relies on an abundant supply of Casimir functions. In some important examples, such as the dynamics of geometrically exact flexible rods, Casimirs have not been found and may not even exist. A method developed to overcome this difficulty is known as the energy– momentum method , which is closely linked to the method of reduction. It uses conserved quantities, namely the energy and momentum map, that are readily available, rather than Casimirs. The energy momentum method (Marsden, Simo, Lewis, and Posbergh [1989], Simo, Posbergh, and Marsden [1990, 1991], Simo, Lewis, and Mars- den [1991], and Lewis and Simo [1990]) involves the augmented Hamil- tonian defined by Hξ(q, p) = H(q, p)− ξ · J(q, p) (1.5.6) where J is the momentum map described in the previous section and ξ may be thought of as a Lagrange multiplier. For the water molecule, J is the angular momentum and ξ is the angular velocity of the relative equilibrium. One sets the first variation of Hξ equal to zero to obtain the relative equi- libria. To ascertain stability, the second variation δ2Hξ is calculated. One is then interested in determining the definiteness of the second variation. Definiteness in this context has to be properly interpreted to take into account the conservation of the momentum map J and the fact that d2Hξ may have zero eigenvalues due to its invariance under a subgroup of the symmetry group. The variations of P and Q must satisfy the linearized an- gular momentum constraint (δq, δp) ∈ ker[DJ(qe, pe)], and must not lie in symmetry directions; only these variations are used to calculate the second variation of the augmented Hamiltonian Hξ. These define the space of ad- missible variations V. The energy momentum method has been applied to the stability of relative equilibria of among others, geometrically exact rods and coupled rigid bodies (Patrick [1989, 1990] and Simo, Posbergh, and Marsden [1990, 1991]). A cornerstone in the development of the energy–momentum method was laid by Routh [1877] and Smale [1970] who studied the stability of relative equilibria of simple mechanical systems. Simple mechanical systems are those whose Hamiltonian may be written as the sum of the potential and 18 1. Introduction kinetic energies. Part of Smale’s work may be viewed as saying that there is a naturally occurring connection called the mechanical connection on the reduction bundle that plays an important role. A connection can be thought of as a generalization of the electromagnetic vector potential. The amended potential Vµ is the potential energy of the system plus a generalization of the potential energy of the centrifugal forces in stationary rotation: Vµ(q) = V (q) + 1 2 µ · I−1(q)µ, (1.5.7) where I is the locked inertia tensor , a generalization of the inertia tensor of the rigid structure obtained by locking all joints in the configuration Q. We will define it precisely in Chapter 3 and compute it for several examples. Smale showed that relative equilibria are critical points of the amended potential Vµ, a result we prove in Chapter 4. The corresponding momentum P need not be zero since the system is typically in motion. The second variation δ2Vµ of Vµ directly yields the stability of the relative equilibria. However, an interesting phenomenon occurs if the space V of admissible variations is split into two specially chosen subspaces VRIG and VINT. In this case the second variation block diagonalizes: δ2Vµ | V × V =  D2Vµ | VRIG × VRIG 0 0 D2Vµ | VINT × VINT  (1.5.8) The space VRIG (rigid variations) is generated by the symmetry group, and VINT are the internal or shape variations. In addition, the whole matrix δ2Hξ block diagonalizes in a very efficient manner as we will see in Chapter 5. This often allows the stability conditions associated with δ2Vµ | V ×V to be recast in terms of a standard eigenvalue problem for the second variation of the amended potential. This splitting, that is, block diagonalization, has more miracles associ- ated with it. In fact, the second variation δ2Hξ and the symplectic structure (and therefore the equations of motion) can be explicitly brought into normal form simultaneously . This result has several interesting implications. In the case of pseudo-rigid bodies (Lewis and Simo [1990]), it reduces the stability problem from an unwieldy 14 × 14 matrix to a relatively simple 3 × 3 subblock on the di- agonal. The block diagonalization procedure enabled Lewis and Simo to solve their problem analytically, whereas without it, a substantial numeri- cal computation would have been necessary. As we shall see in Chapter 8, the presence of discrete symmetries (as for the water molecule and the pseudo-rigid bodies) gives further, or refined, 1.6 Geometric Phases 19 subblocking properties in the second variation of δ2Hξ and δ2Vµ and the symplectic form. In general, this diagonalization explicitly separates the rotational and internal modes, a result which is important not only in rotating and elastic fluid systems, but also in molecular dynamics and robotics. Similar simpli- fications are expected in the analysis of other problems to be tackled using the energy momentum method. 1.6 Geometric Phases The application of the methods described above is still in its infancy, but the previous example indicates the power of reduction and suggests that the energy–momentum method will be applied to dynamic problems in many fields, including chemistry, quantum and classical physics, and engineering. Apart from the computational simplification afforded by reduction, reduc- tion also permits us to put into a mechanical context a concept known as the geometric phase , or holonomy . An example in which holonomy occurs is the Foucault pendulum. During a single rotation of the earth, the plane of the pendulum’s oscillations is shifted by an angle that depends on the latitude of the pendulum’s location. Specifically if a pendulum located at co-latitude (i.e., the polar angle) α is swinging in a plane, then after twenty-four hours, the plane of its oscillations will have shifted by an angle 2pi cosα. This holonomy is (in a non-obvious way) a result of parallel translation: if an orthonormal coordinate frame undergoes parallel transport along a line of co-latitude α, then after one revolution the frame will have rotated by an amount equal to the phase shift of the Foucault pendulum (see Figure 1.6.1). Geometrically, the holonomy of the Foucault pendulum is equal to the solid angle swept out by the pendulum’s axis during one rotation of the earth. Thus a pendulum at the north pole of the earth will experience a holonomy of 2pi. If you imagine parallel transporting a vector around a small loop near the north pole, it is clear that one gets an answer close to 2pi, which agrees with what the pendulum experiences. On the other hand, a pendulum on the earth’s equator experiences no holonomy. A less familiar example of holonomy was presented by Hannay [1985] and discussed further by Berry [1985]. Consider a frictionless, non-circular, planar hoop of wire on which is placed a small bead. The bead is set in motion and allowed to slide along the wire at a constant speed (see Figure 1.6.2). (We will need the notation in this figure only later in Chapter 6.) Clearly the bead will return to its initial position after, say, τ seconds, and will continue to return every τ seconds after that. Allow the bead to make many revolutions along the circuit, but for a fixed amount of total time, say T . Suppose that the wire hoop is slowly rotated in its plane by 20 1. Introduction cut and unroll cone parallel translate fram e along a line of latitude Figure 1.6.1. The parallel transport of a coordinate frame along a curved surface. S Figure 1.6.2. A bead sliding on a planar, non-circular hoop of area A and length L. The bead slides around the hoop at constant speed with period τ and is allowed to revolve for time T . 360 degrees while the bead is in motion for exactly the same total length of time T . At the end of the rotation, the bead is not in the location where we might expect it, but instead will be found at a shifted position that is determined by the shape of the hoop. In fact, the shift in position depends only on the length of the hoop, L, and on the area it encloses, A. The shift is given by 8pi2A/L2 as an angle, or by 4piA/L as length. (See §6.6 for a derivation of these formulas.) To be completely concrete, if the bead’s initial position is marked with a tick and if the time of rotation is a multiple of the bead’s period, then at the end of rotation the bead is found approximately 4piA/L units from its initial position. This is shown in Figure 1.6.3. Note that if the hoop is circular then the angular shift is 2pi. Let us indicate how holonomy is linked to the reduction process by returning to our rigid body example. The rotational motion of a rigid body can be described as a geodesic (with respect to the inertia tensor regarded as a metric) on the 1.6 Geometric Phases 21 Figure 1.6.3. The hoop is slowly rotated in the plane through 360 degrees. After one rotation, the bead is located 4piA/L units behind where it would have been, had the rotation not occurred. manifold SO(3). As mentioned earlier, for each angular momentum vector µ, the reduced space Pµ can be identified with the two-sphere of radius ‖µ‖. This construction corresponds to the Hopf fibration which describes the three-sphere S3 as a nontrivial circle bundle over S2. In our example, S3 (or rather S3/Z2 ∼= J−1(µ)) is the subset of phase space which is mapped to µ under the reduction process. Suppose we are given a trajectory Π(t) on Pµ that has period T and energy E. Following Montgomery [1991] and Marsden, Montgomery, and Ratiu [1990] we shall show in §6.4 that after time T the rigid body has rotated in physical 3-space about the axis µ by an angle (modulo 2pi) ∆θ = −Λ + 2ET‖µ‖ . (1.6.1) Here Λ is the solid angle subtended by the closed curve Π(t) on the sphere S2 and is oriented according to the right hand rule. The approximate phase formula ∆θ ∼= 8pi2A/L2 for the ball in the hoop is derived by the classical techniques of averaging and the variation of constants formula. However, formula (1.6.1) is exact . (In Whittaker [1959], (1.6.1) is expressed as a complicated quotient of theta functions!) An interesting feature of (1.6.1) is the manner in which ∆θ is split into two parts. The term Λ is purely geometric and so is called the geometric phase . It does not depend on the energy of the system or the period of motion, but rather on the fraction of the surface area of the sphere Pµ that is enclosed by the trajectory Π(t). The second term in (1.6.1) is known as the dynamic phase and depends explicitly on the system’s energy and the period of the reduced trajectory. Geometrically we can picture the rigid body as tracing out a path in its phase space. More precisely, conservation of angular momentum im- plies that the path lies in the submanifold consisting of all points that are 22 1. Introduction mapped onto µ by the reduction process. As Figure 1.2.1 shows, almost every trajectory on the reduced space is periodic, but this does not imply that the original path was periodic, as is shown in Figure 1.6.4. The differ- ence between the true trajectory and a periodic trajectory is given by the holonomy plus the dynamic phase. P μ dynam ic phase geom etric phase true trajectory horizontal lift reduced trajectory D Figure 1.6.4. Holonomy for the rigid body. As the body completes one period in the reduced phase space Pµ, the body’s true configuration does not return to its original value. The phase difference is equal to the influence of a dynamic phase which takes into account the body’s energy, and a geometric phase which depends only on the area of Pµ enclosed by the reduced trajectory. It is possible to observe the holonomy of a rigid body with a simple experiment. Put a rubber band around a book so that the cover will not open. (A “tall”, thin book works best.) With the front cover pointing up, gently toss the book in the air so that it rotates about its middle axis (see Figure 1.6.5). Catch the book after a single rotation and you will find that it has also rotated by 180 degrees about its long axis—that is, the front cover is now facing the floor! This particular phenomena is not literally covered by Montgomery’s for- mula since we are working close to the homoclinic orbit and in this limit ∆θ → +∞ due to the limiting steady rotations. Thus, “catching” the book plays a role. For an analysis from another point of view, see Ashbaugh, Chicone, and Cushman [1991]. There are other everyday occurrences that demonstrate holonomy. For example, a falling cat usually manages to land upright if released upside down from complete rest, that is, with total angular momentum zero. This ability has motivated several investigations in physiology as well as dy- 1.6 Geometric Phases 23 Figure 1.6.5. A book tossed in the air about an axis that is close to the middle (unstable) axis experiences a holonomy of approximately 180 degrees about its long axis when caught after one revolution. namics and more recently has been analyzed by Montgomery [1990] with an emphasis on how the cat (or, more generally, a deformable body) can efficiently readjust its orientation by changing its shape. By “efficiently”, we mean that the reorientation minimizes some function—for example the total energy expended. In other words, one has a problem in optimal con- trol . Montgomery’s results characterize the deformations that allow a cat to reorient itself without violating conservation of angular momentum. In his analysis, Montgomery casts the falling cat problem into the language of principal bundles. Let the shape of a cat refer to the location of the cat’s body parts relative to each other, but without regard to the cat’s orientation in space. Let the configuration of a cat refer both to the cat’s shape and to its orientation with respect to some fixed reference frame. More precisely, if Q is the configuration space and G is the group of rigid motions, then Q/G is the shape space . If the cat is completely rigid then it will always have the same shape, but we can give it a different configuration by rotating it through, say 180 degrees about some axis. If we require that the cat has the same shape at the end of its fall as it had at the beginning, then the cat problem may be formulated as follows: Given an initial configuration, what is the most efficient way for a cat to achieve a desired final configuration if the final shape is required to be the same as the initial shape? If we think of the cat as tracing out some path in configuration space during its fall, the projection of this path onto the shape space results in a trajectory in the shape space, and the requirement that the cat’s initial and final shapes are the same means that the trajectory is a closed loop. Furthermore, if we want to know the most efficient configuration path that satisfies the initial and final conditions, then we want to find the shortest path with respect to a metric induced by the function we wish to minimize. It turns out that the solution of a falling cat problem is closely related to Wong’s equations that 24 1. Introduction describe the motion of a colored particle in a Yang–Mills field (Montgomery [1990], Shapere and Wilczeck [1989]). We will come back to these points in Chapter 7. The examples above indicate that holonomic occurrences are not rare. In fact, Shapere and Wilczek showed that aquatic microorganisms use holon- omy as a form of propulsion. Because these organisms are so small, the environment in which they live is extremely viscous to them. The apparent viscosity is so great, in fact, that they are unable to swim by conventional stroking motions, just as a person trapped in a tar pit would be unable to swim to safety. These microorganisms surmount their locomotion dif- ficulties, however, by moving their “tails” or changing their shapes in a topologically nontrivial way that induces a holonomy and allows them to move forward through their environment. There are probably many conse- quences and applications of “holonomy drive” that remain to be discovered. Yang and Krishnaparasd [1990] have provided an example of holonomy drive for coupled rigid bodies linked together with pivot joints as shown in Figure 1.6.5. (For simplicity, the bodies are represented as rigid rods.) This form of linkage permits the rods to freely rotate with respect of each other, and we assume that the system is not subjected to external forces or torques although torques will exist in the joints as the assemblage rotates. By our assumptions, angular momentum is conserved in this system. Yet, even if the total angular momentum is zero, a turn of the crank (as indicated in Figure 1.6.6) returns the system to its initial shape but creates a holonomy that rotates the system’s configuration. See Thurston and Weeks [1984] for some relationships between linkages and the theory of 3-manifolds (they do not study dynamics, however). Brockett [1987, 1989] studies the use of holonomy in micromotors. overall phase rotation of the assem blage crank Figure 1.6.6. Rigid rods linked by pivot joints. As the “crank” traces out the path shown, the assemblage experiences a holonomy resulting in a clockwise shift in its configuration. Holonomy also comes up in the field of magnetic resonance imaging (MRI) and spectroscopy. Berry’s work shows that if a quantum system 1.6 Geometric Phases 25 experiences a slow (adiabatic) cyclic change, then there will be a shift in the phase of the system’s wave function. This is a quantum analogue to the bead on a hoop problem discussed above. This work has been verified by several independent experiments; the implications of this result to MRI and spectroscopy are still being investigated. For a review of the appli- cations of the geometric phase to the fields of spectroscopy, field theory, and solid-state physics, see Zwanziger, Koenig, and Pines [1990] and the bibliography therein. Another possible application of holonomy drive is to the somersaulting robot. Due to the finite precision response of motors and actuators, a slight error in the robot’s initial angular momentum can result in an unsatisfac- tory landing as the robot attempts a flip. Yet, in spite of the challenges, Hodgins and Raibert [1990] report that the robot can execute 90 percent of the flips successfully. Montgomery, Raibert, and Li [1991] are asking whether a robot can use holonomy to improve this rate of success. To do this, they reformulate the falling cat problem as a problem in feedback con- trol: the cat must use information gained by its senses in order to determine how to twist and turn its body so that it successfully lands on its feet. It is possible that the same technique used by cats can be implemented in a robot that also wants to complete a flip in mid-air. Imagine a robot installed with sensors so that as it begins its somersault it measures its momenta (linear and angular) and quickly calculates its final landing po- sition. If the calculated final configuration is different from the intended final configuration, then the robot waves mechanical arms and legs while entirely in the air to create a holonomy that equals the difference between the two configurations. If “holonomy drive” can be used to control a mechanical structure, then there may be implications for future satellites like a space telescope. Sup- pose the telescope initially has zero angular momentum (with respect to its orbital frame), and suppose it needs to be turned 180 degrees. One way to do this is to fire a small jet that would give it angular momentum, then, when the turn is nearly complete, to fire a second jet that acts as a brake to exactly cancel the acquired angular momentum. As in the somersaulting robot, however, errors are bound to occur, and the process of returning the telescope to (approximately) zero angular momentum may be a long process. It would seem to be more desirable to turn it while constantly pre- serving zero angular momentum. The falling cat performs this very trick. A telescope can mimic this with internal momentum wheels or with flexi- ble joints. This brings us to the area of control theory and its relation to the present ideas. Bloch, Krishnaprasad, Marsden, and Sanchez de Alvarez [1992] represents one step in this direction. We shall also discuss their result in Chapter 7. 26 1. Introduction 1.7 The Rotation Group and the Poincare´ Sphere The rotation group SO(3), consisting of all 3× 3 orthogonal matrices with determinant one, plays an important role for problems of interest in this book, and so one should try to understand it a little more deeply. As a first try, one can contemplate Euler’s theorem, which states that every rotation in R3 is a rotation through some angle about some axis. While true, this can be misleading if not used with care. For example, it suggests that we can identify the set of rotations with the set consisting of all unit vectors in R3 (the axes) and numbers between 0 and 2pi (the angles); that is, with the set S2 × S1. However, this is false for reasons that involve some basic topology. Thus, a better approach is needed. One method for gaining deeper insight is to realize SO(3) as SU(2)/Z2, where SU(2) is the group of 2×2 complex unitary matrices of determinant 1, using quaternions and Pauli spin matrices; see Abraham and Marsden [1978, pp. 273–4], for the precise statement. This approach also shows that the group SU(2) is diffeomorphic to the set of unit quaternions, the three sphere S3 in R4. The Hopf fibration is the map of S3 to S2 defined, using the above approach to the rotation group, as follows. First map a point w ∈ S3 to its equivalence class A = [w] ∈ SO(3) and then map this point to Ak, where k is the standard unit vector in R3 (or any other fixed unit vector in R3). Closely related to the above description of the rotation group, but in fact a little more straightforward, is Poincare´’s representation of SO(3) as the unit circle bundle T1S2 of the two sphere S2. This comes about as follows. Elements of SO(3) can be identified with oriented orthonormal frames in R3; that is, with triples of orthonormal vectors (n,m,n ×m). Such triples are in one to one correspondence with the points in T1S2 by (n,m,n×m)↔ (n,m) where n is the base point in S2 and m is regarded as a vector tangent to S2 at the point n (see Figure 1.7.1). Poincare´’s representation shows that SO(3) cannot be written as S2×S1 (i.e., one cannot globally and in a unique and singularity free way, write rotations using an axis and an angle, despite Euler’s theorem). This is because of the (topological) fact that every vector field on S2 must vanish somewhere. Thus, we see that this is closely related to the nontriviality of the Hopf fibration. Not only does Poincare´’s representation show that SO(3) is topologi- cally non-trivial, but this representation is useful in mechanics. In fact, Poincare´’s representation of SO(3) as the unit circle bundle T1S2 is the stepping stone in the formulation of optimal singularity free parameteriza- tions for geometrically exact plates and shells. These ideas have been ex- ploited within a numerical analysis context in Simo, Fox, and Rifai [1990] for statics and Simo, Fox, and Rifai [1991] for dynamics. Also, this represen- 1.7 The Rotation Group and the Poincare´ Sphere 27 Figure 1.7.1. The rotation group is diffeomorphic to the unit tangent bundle to the two sphere. tation is helpful in studying the problem of reorienting a rigid body using internal momentum wheels. We shall treat some aspects of these topics in the course of the lectures. 28 1. Introduction Page 29 2 A Crash Course in Geometric Mechanics We now set out some of the notation and terminology that is used in sub- sequent chapters. The reader is referred to one of the standard books, such as Abraham and Marsden [1978], Arnold [1989], Guillemin and Sternberg [1984] and Marsden and Ratiu [1999] for proofs omitted here.1 2.1 Symplectic and Poisson Manifolds 2.1.1 Definition. Let P be a manifold and let F(P ) denote the set of smooth real-valued functions on P . Consider a given bracket operation de- noted { , } : F(P )×F(P )→ F(P ). The pair (P, { , }) is called a Poisson manifold if { , } satisfies PB1. bilinearity {f, g} is bilinear in f and g, PB2. anticommutativity {f, g} = −{g, f}, PB3. Jacobi’s identity {{f, g}, h}+ {{h, f}, g}+ {{g, h}, f} = 0, PB4. Leibnitz’ rule {fg, h} = f{g, h}+ g{f, h}. Conditions PB1–PB3 make (F(P ), { , }) into a Lie algebra. If (P, { , }) is a Poisson manifold, then because of PB1 and PB4, there is a tensor B 1We thank Hans Peter Kruse for providing a helpful draft of the notes for this lecture. 30 2. A Crash Course in Geometric Mechanics on P , assigning to each z ∈ P a linear map B(z) : T ∗z P → TzP such that {f, g}(z) = 〈B(z) · df(z),dg(z)〉. (2.1.1) Here, 〈 , 〉 denotes the natural pairing between vectors and covectors. Be- cause of PB2, B(z) is antisymmetric. Letting zI , I = 1, . . . ,M denote coordinates on P , (2.1.1) becomes {f, g} = BIJ ∂f ∂zI ∂g ∂zJ . (2.1.2) Antisymmetry means BIJ = −BJI and Jacobi’s identity reads BLI ∂BJK ∂zL +BLJ ∂BKI ∂zL +BLK ∂BIJ ∂zL = 0. (2.1.3) 2.1.2 Definition. Let (P1, { , }1) and (P2, { , }2) be Poisson manifolds. A mapping ϕ : P1 → P2 is called Poisson if for all f, h ∈ F(P2), we have {f, h}2 ◦ ϕ = {f ◦ ϕ, h ◦ ϕ}1. (2.1.4) 2.1.3 Definition. Let P be a manifold and Ω a 2-form on P . The pair (P,Ω) is called a symplectic manifold if Ω satisfies S1. dΩ = 0 (i.e., Ω is closed) and S2. Ω is nondegenerate. 2.1.4 Definition. Let (P,Ω) be a symplectic manifold and let f ∈ F(P ). Let Xf be the unique vector field on P satisfying Ωz(Xf (z), v) = df(z) · v for all v ∈ TzP. (2.1.5) We call Xf the Hamiltonian vector field of f . Hamilton’s equations are the differential equations on P given by z˙ = Xf (z). (2.1.6) If (P,Ω) is a symplectic manifold, define the Poisson bracket operation {·, ·} : F(P )×F(P )→ F(P ) by {f, g} = Ω(Xf , Xg). (2.1.7) The construction (2.1.7) makes (P, { , }) into a Poisson manifold. In other words, 2.1.5 Proposition. Every symplectic manifold is Poisson. The converse is not true; for example the zero bracket makes any manifold Poisson. In §2.4 we shall see some non-trivial examples of Poisson brackets that are not symplectic, such as Lie–Poisson structures on duals of Lie algebras. Hamiltonian vector fields are defined on Poisson manifolds as follows. 2.2 The Flow of a Hamiltonian Vector Field 31 2.1.6 Definition. Let (P, { , }) be a Poisson manifold and let f ∈ F(P ). Define Xf to be the unique vector field on P satisfying Xf [k] : = 〈dk,Xf 〉 = {k, f} for all k ∈ F(P ). We call Xf the Hamiltonian vector field of f . A check of the definitions shows that in the symplectic case, the Def- initions 2.1.4 and 2.1.6 of Hamiltonian vector fields coincide. If (P, { , }) is a Poisson manifold, there are therefore three equivalent ways to write Hamilton’s equations for H ∈ F(P ): (i) z˙ = XH(z), (ii) f˙ = df(z) ·XH(z) for all f ∈ F(P ), and (iii) f˙ = {f,H} for all f ∈ F(P ). 2.2 The Flow of a Hamiltonian Vector Field Hamilton’s equations described in the abstract setting of the last section are very general. They include not only what one normally thinks of as Hamil- ton’s canonical equations in classical mechanics, but also the equations of elasticity, fluids, electromagnetism, Schro¨dinger’s equation in quantum me- chanics to mention a few. Despite this generality, as we shall see, the theory has a rich structure. LetH ∈ F(P ) where P is a Poisson manifold. Let ϕt be the flow of Hamil- ton’s equations; thus, ϕt(z) is the integral curve of z˙ = XH(z) starting at z. (If the flow is not complete, restrict attention to its domain of definition.) There are two basic facts about Hamiltonian flows (ignoring functional analytic technicalities in the infinite dimensional case—see Chernoff and Marsden [1974]). 2.2.1 Proposition. The following hold for Hamiltonian systems on Pois- son manifolds: (i) each ϕt is a Poisson map, (ii) H ◦ ϕt = H (conservation of energy). The first part of this proposition is true even if H is a time dependent Hamiltonian, while the second part is true only when H is independent of time. 32 2. A Crash Course in Geometric Mechanics 2.3 Cotangent Bundles Let Q be a given manifold (usually the configuration space of a mechanical system) and T ∗Q be its cotangent bundle. Coordinates qi on Q induce coordinates (qi, pj) on T ∗Q, called the canonical cotangent coordinates of T ∗Q. 2.3.1 Proposition. There is a unique 1-form Θ on T ∗Q such that in any choice of canonical cotangent coordinates, Θ = pidqi; (2.3.1) Θ is called the canonical 1-form . We define the canonical 2-form Ω by Ω = −dΘ = dqi ∧ dpi (a sum on i is understood). (2.3.2) In infinite dimensions, one needs to use an intrinsic definition of Θ, and there are many such; one of these is the identity β∗Θ = β for β : Q→ T ∗Q any one form. Another is Θ(wαq ) = 〈αq, TpiQ · wαq 〉, where αq ∈ T ∗qQ,wαq ∈ Tαq (T ∗Q) and where piQ : T ∗Q→ Q is the cotan- gent bundle projection. 2.3.2 Proposition. (T ∗Q,Ω) is a symplectic manifold. In cotangent bundle coordinates (naturally induced from coordinates on Q), the Poisson brackets on T ∗Q have the classical form {f, g} = ∂f ∂qi ∂g ∂pi − ∂g ∂qi ∂f ∂pi , (2.3.3) where summation on repeated indices is understood. 2.3.3 Theorem (Darboux’ Theorem). Every symplectic manifold locally looks like T ∗Q; in other words, on every finite dimensional symplectic man- ifold, there are local coordinates in which Ω has the form (2.3.2). (See Marsden [1981] and Olver [1988] for a discussion of the infinite dimensional case.) Hamilton’s equations in these canonical coordinates have the classical form q˙i = ∂H ∂pi , p˙i = −∂H ∂qi , as one can readily check. 2.4 Lagrangian Mechanics 33 The local structure of Poisson manifolds is more complex than the sym- plectic case. However, Kirillov [1976] has shown that every Poisson manifold is the union of symplectic leaves; to compute the bracket of two functions in P , one does it leaf-wise . In other words, to calculate the bracket of f and g at z ∈ P , select the symplectic leaf Sz through z, and evaluate the bracket of f |Sz and g|Sz at z. We shall see a specific case of this picture shortly. 2.4 Lagrangian Mechanics Let Q be a manifold and TQ its tangent bundle. Coordinates qi on Q induce coordinates (qi, q˙i) on TQ, called tangent coordinates. A mapping L : TQ → R is called a Lagrangian . It is often the case that L has the form L = K − V where K(v) = 12 〈〈v, v〉〉 is the kinetic energy associated to a given Riemannian metric 〈〈 , 〉〉, and where V : Q → R is a potential energy function. 2.4.1 Definition. Hamilton’s principle singles out particular curves q(t) by the condition δ ∫ a b L(q(t), q˙(t))dt = 0, (2.4.1) where the variation is over smooth curves in Q with fixed endpoints. Note that (2.4.1) is unchanged if we replace the integrand by L(q, q˙) − d dtS(q, t) for any function S(q, t). This reflects the gauge invariance of classical mechanics and is closely related to Hamilton–Jacobi theory. If one prefers, the action principle states that the map I defined by I(q(·)) = ∫ b a L(q(t), q˙(t))dt from the space of curves with prescribed end- points in Q to R has a critical point at the curve in question. In any case, a basic and elementary result of the calculus of variations, whose proof was sketched in §1.2, is: 2.4.2 Proposition. Hamilton’s principle for a curve q(t) is equivalent to the condition that q(t) satisfies the Euler–Lagrange equations d dt ∂L ∂q˙i − ∂L ∂qi = 0. (2.4.2) Since Hamilton’s principle is intrinsic (independent of coordinates), the Euler–Lagrange equations must be as well. It is interesting to do so—see, for example, Marsden and Ratiu [1999] for an exposition of this. Another interesting fact is that if one keeps track of the boundary con- ditions in Hamilton’s principle, they essentially define the canonical one form, pidqi. In fact, this can be carried further and used as a completely 34 2. A Crash Course in Geometric Mechanics different way of showing that the flow of the Euler–Lagrange equations are symplectic. Namely, one considers the action function on solution curves (q(t), q˙(t)) emanating from given initial conditions (q0, q˙0), thought of as a function of time and the initial conditions: S(q0, q˙0, t) = ∫ t 0 L(q(s), q˙(s), s) ds Interestingly, the identity d2S = 0 gives the symplecticity condition, an argument due to Marsden, Patrick and Shkoller [1998], which we refer to for details and multisymplectic generalizations. 2.4.3 Definition. Let L be a Lagrangian on TQ and let FL : TQ→ T ∗Q be defined (in coordinates) by (qi, q˙j) 7→ (qi, pj) where pj = ∂L/∂q˙j. We call FL the fiber derivative. (Intrinsically, FL differentiates L in the fiber direction.) A Lagrangian L is called hyperregular if FL is a diffeomorphism. If L is a hyperregular Lagrangian, we define the corresponding Hamiltonian by H(qi, pj) = piq˙i − L. The change of data from L on TQ to H on T ∗Q is called the Legendre transform . One checks that the Euler–Lagrange equations for L are equivalent to Hamilton’s equations for H. In a relativistic context one finds that the two conditions pj = ∂L/∂q˙j and H = piq˙i − L, defining the Legendre transform, fit together as the spatial and temporal components of a single object. Suffice it to say that the formalism developed here is useful in the context of relativistic fields. 2.5 Lie–Poisson Structures and the Rigid Body Not every Poisson manifold is symplectic. For example, a large class of non- symplectic Poisson manifolds is the class of Lie–Poisson manifolds, which we now define. Let G be a Lie group and g = TeG its Lie algebra with [ , ] : g× g→ g the associated Lie bracket. 2.5.1 Proposition. The dual space g∗ is a Poisson manifold with either of the two brackets {f, k}±(µ) = ± 〈 µ, [ δf δµ , δk δµ ]〉 . (2.5.1) 2.5 Lie–Poisson Structures and the Rigid Body 35 Here g is identified with g∗∗ in the sense that δf/δµ ∈ g is defined by 〈ν, δf/δµ〉 = df(µ) · ν for ν ∈ g∗, where d denotes the derivative. (In the infinite dimensional case one needs to worry about the existence of δf/δµ; in this context, methods like the Hahn–Banach theorem are not always appropriate!) The notation δf/δµ is used to conform to the functional derivative notation in classical field theory. In coordinates, (ξ1, . . . , ξm) on g and corresponding dual coordinates (µ1, . . . , µm) on g∗, the Lie–Poisson bracket (2.5.1) is {f, k}±(µ) = ±µaCabc ∂f ∂µb ∂k ∂µc ; (2.5.2) here Cabc are the structure constants of g defined by [ea, eb] = C c abec, where (e1, . . . , em) is the coordinate basis of g and where, for ξ ∈ g, we write ξ = ξaea, and for µ ∈ g∗, µ = µaea, where (e1, . . . , em) is the dual basis. Formula (2.5.2) appears explicitly in Lie [1890, Section 75]. Which sign to take in (2.5.2) is determined by understanding Lie– Poisson reduction , which can be summarized as follows. Let λ : T ∗G→ g∗ be defined by pg 7→ (TeLg)∗pg ∈ T ∗eG ∼= g∗ (2.5.3) and ρ : T ∗G→ g∗ be defined by pg 7→ (TeRg)∗pg ∈ T ∗eG ∼= g∗. (2.5.4) Then λ is a Poisson map if one takes the (−) Lie–Poisson structure on g∗ and ρ is a Poisson map if one takes the (+) Lie–Poisson structure on g∗. Every left invariant Hamiltonian and Hamiltonian vector field is mapped by λ to a Hamiltonian and Hamiltonian vector field on g∗. There is a similar statement for right invariant systems on T ∗G. One says that the original system on T ∗G has been reduced to g∗. The reason λ and ρ are both Poisson maps is perhaps best understood by observing that they are both equivariant momentum maps generated by the action of G on itself by right and left translations, respectively. We take up this topic in §2.7. We saw in Chapter 1 that the Euler equations of motion for rigid body dynamics are given by Π˙ = Π× Ω, (2.5.5) where Π = IΩ is the body angular momentum and Ω is the body angu- lar velocity. Euler’s equations are Hamiltonian relative to a Lie–Poisson structure. To see this, take G = SO(3) to be the configuration space. Then g ∼= (R3,×) and we identify g ∼= g∗. The corresponding Lie–Poisson struc- ture on R3 is given by {f, k}(Π) = −Π · (∇f ×∇k). (2.5.6) For the rigid body one chooses the minus sign in the Lie–Poisson bracket. This is because the rigid body Lagrangian (and hence Hamiltonian) is left invariant and so its dynamics pushes to g∗ by the map λ in (2.5.3). 36 2. A Crash Course in Geometric Mechanics Starting with the kinetic energy Hamiltonian derived in Chapter 1, we directly obtain the formula H(Π) = 12Π · (I−1Π), the kinetic energy of the rigid body. One verifies from the chain rule and properties of the triple product that: 2.5.2 Proposition. Euler’s equations are equivalent to the following equation for all f ∈ F(R3): f˙ = {f,H}. (2.5.7) 2.5.3 Definition. Let (P, { , }) be a Poisson manifold. A function C ∈ F(P ) satisfying {C, f} = 0 for all f ∈ F(P ) (2.5.8) is called a Casimir function . A crucial difference between symplectic manifolds and Poisson manifolds is this: On symplectic manifolds, the only Casimir functions are the con- stant functions (assuming P is connected). On the other hand, on Poisson manifolds there is often a large supply of Casimir functions. In the case of the rigid body, every function C : R3 → R of the form C(Π) = Φ(‖Π‖2) (2.5.9) where Φ : R → R is a differentiable function, is a Casimir function, as we noted in Chapter 1. Casimir functions are constants of the motion for any Hamiltonian since C˙ = {C,H} = 0 for any H. In particular, for the rigid body, ‖Π‖2 is a constant of the motion—this is the invariant sphere we saw in Chapter 1. There is an intimate relation between Casimirs and symmetry generated conserved quantities, or momentum maps, which we study in §2.7. The maps λ and ρ induce Poisson isomorphisms between (T ∗G)/G and g∗ (with the (−) and (+) brackets respectively) and this is a special instance of Poisson reduction, as we will see in §2.8. The following result is one useful way of formulating the general relation between T ∗G and g∗. We treat the left invariant case to be specific. Of course, the right invariant case is similar. 2.5.4 Theorem. Let G be a Lie group and H : T ∗G → R be a left invariant Hamiltonian. Let h : g∗ → R be the restriction of H to the identity. For a curve p(t) ∈ T ∗g(t)G, let µ(t) = (T ∗g(t)L) · p(t) = λ(p(t)) be the induced curve in g∗. Assume that g˙ = ∂H/∂p ∈ TgG. Then the following are equivalent: (i) p(t) is an integral curve of XH ; that is, Hamilton’s equations on T ∗G hold, (ii) for any F ∈ F(T ∗G), F˙ = {F,H}, where { , } is the canonical bracket on T ∗G, 2.6 The Euler–Poincare´ Equations 37 (iii) µ(t) satisfies the Lie–Poisson equations dµ dt = ad∗δh/δµµ (2.5.10) where adξ : g→ g is defined by adξη = [ξ, η] and ad∗ξ is its dual, that is, µ˙a = Cdba δh δµb µd, (2.5.11) (iv) for any f ∈ F(g∗), we have f˙ = {f, h}− (2.5.12) where { , }− is the minus Lie–Poisson bracket. We now make some remarks about the proof. First of all, the equivalence of (i) and (ii) is general for any cotangent bundle, as we have already noted. Next, the equivalence of (ii) and (iv) follows directly from the fact that λ is a Poisson map (as we have mentioned, this follows from the fact that λ is a momentum map; see Proposition 2.7.6 below) and H = h ◦ λ. Finally, we establish the equivalence of (iii) and (iv). Indeed, f˙ = {f, h}− means〈 µ˙, δf δµ 〉 = − 〈 µ, [ δf δµ , δh δµ ]〉 = 〈 µ, adδh/δµ δf δµ 〉 = 〈 ad∗δh/δµµ, δf δµ 〉 . Since f is arbitrary, this is equivalent to iii. 2.6 The Euler–Poincare´ Equations In §1.3 we saw that for the rigid body, there is an analogue of the above theorem on SO(3) and so(3) using the Euler–Lagrange equations and the variational principle as a starting point. We now generalize this to an arbi- trary Lie group and make the direct link with the Lie–Poisson equations. 2.6.1 Theorem. Let G be a Lie group and L : TG→ R a left invariant Lagrangian. Let l : g → R be its restriction to the identity. For a curve g(t) ∈ G, let ξ(t) = g(t)−1 · g˙(t); that is ξ(t) = Tg(t)Lg(t)−1 g˙(t). Then the following are equivalent 38 2. A Crash Course in Geometric Mechanics (i) g(t) satisfies the Euler–Lagrange equations for L on G, (ii) the variational principle δ ∫ L(g(t), g˙(t))dt = 0 (2.6.1) holds, for variations with fixed endpoints, (iii) the Euler–Poincare´ equations hold: d dt δl δξ = ad∗ξ δl δξ , (2.6.2) (iv) the variational principle δ ∫ l(ξ(t))dt = 0 (2.6.3) holds on g, using variations of the form δξ = η˙ + [ξ, η], (2.6.4) where η vanishes at the endpoints. Let us discuss the main ideas of the proof. First of all, the equivalence of (i) and (ii) holds on the tangent bundle of any configuration manifold Q, as we have seen, secondly, (ii) and (iv) are equivalent. To see this, one needs to compute the variations δξ induced on ξ = g−1g˙ = TLg−1 g˙ by a variation of g. To calculate this, we need to differentiate g−1g˙ in the direction of a variation δg. If δg = dg/d at = 0, where g is extended to a curve g, then, δξ = d d ( g−1 d dt g )∣∣∣∣ =0 while if η = g−1δg, then η˙ = d dt ( g−1 d d g )∣∣∣∣ =0 . The difference δξ − η˙ is the commutator, [ξ, η]. This argument is fine for matrix groups, but takes a little more work to make precise for general Lie groups. See Bloch, Krishnaprasad, Marsden, and Ratiu [1996] for the general case. Thus, (ii) and (iv) are equivalent. To complete the proof, we show the equivalence of (iii) and (iv). Indeed, using the definitions and integrating by parts, δ ∫ l(ξ)dt = ∫ δl δξ δξ dt = ∫ δl δξ (η˙ + adξη)dt = ∫ [ − d dt ( δl δξ ) + ad∗ξ δl δξ ] η dt 2.7 Momentum Maps 39 so the result follows. Generalizing what we saw directly in the rigid body, one can check di- rectly from the Euler–Poincare´ equations that conservation of spatial an- gular momentum holds: d dt pi = 0 (2.6.5) where pi is defined by pi = Ad∗g δl δξ . (2.6.6) Since the Euler–Lagrange and Hamilton equations on TQ and T ∗Q are equivalent, it follows that the Lie–Poisson and Euler–Poincare´ equations are also equivalent under a regularity hypothesis. To see this directly, we make the following Legendre transformation from g to g∗: µ = δl δξ , h(µ) = 〈µ, ξ〉 − l(ξ). Note that δh δµ = ξ + 〈 µ, δξ δµ 〉 − 〈 δl δξ , δξ δµ 〉 = ξ and so it is now clear that (2.5.10) and (2.6.2) are equivalent. 2.7 Momentum Maps Let G be a Lie group and P be a Poisson manifold, such that G acts on P by Poisson maps (in this case the action is called a Poisson action). Denote the corresponding infinitesimal action of g on P by ξ 7→ ξP , a map of g to X(P ), the space of vector fields on P . We write the action of g ∈ G on z ∈ P as simply gz; the vector field ξP is obtained at z by differentiating gz with respect to g in the direction ξ at g = e. Explicitly, ξP (z) = d d [exp(ξ) · z] ∣∣∣∣ =0 . 2.7.1 Definition. A map J : P → g∗ is called a momentum map if X〈J,ξ〉 = ξP for each ξ ∈ g, where 〈J, ξ〉(z) = 〈J(z), ξ〉. 2.7.2 Theorem (Noether’s Theorem). If H is a G invariant Hamiltonian on P , then J is conserved on the trajectories of the Hamiltonian vector field XH . Proof. Differentiating the invariance condition H(gz) = H(z) with re- spect to g ∈ G for fixed z ∈ P , we get dH(z)·ξP (z) = 0 and so {H, 〈J, ξ〉} = 0 which by antisymmetry gives d〈J, ξ〉 ·XH = 0 and so 〈J, ξ〉 is conserved on the trajectories of XH for every ξ in G. 40 2. A Crash Course in Geometric Mechanics Turning to the construction of momentum maps, let Q be a manifold and let G act on Q. This action induces an action of G on T ∗Q by cotangent lift—that is, we take the transpose inverse of the tangent lift. The action of G on T ∗Q is always symplectic and therefore Poisson. 2.7.3 Theorem. A momentum map for a cotangent lifted action is given by J : T ∗Q→ g∗ defined by 〈J, ξ〉(pq) = 〈pq, ξQ(q)〉. (2.7.1) In canonical coordinates, we write pq = (qi, pj) and define the action functions Kia by (ξQ) i = Kia(q)ξ a. Then 〈J, ξ〉(pq) = piKia(q)ξa (2.7.2) and therefore Ja = piKia(q). (2.7.3) Recall that by differentiating the conjugation operation h 7→ ghg−1 at the identity, one gets the adjoint action of G on g. Taking its dual produces the coadjoint action of G on g∗. 2.7.4 Proposition. The momentum map for cotangent lifted actions is equivariant , that is, the diagram in Figure 2.7.1 commutes. T ∗Q g∗ T ∗Q g∗ J J G-action on T ∗Q coadjoint action - - ? ? Figure 2.7.1. Equivariance of the momentum map. 2.7.5 Proposition. Equivariance implies infinitesimal equivariance, which can be stated as the classical commutation relations: {〈J, ξ〉, 〈J, η〉} = 〈J, [ξ, η]〉. 2.7.6 Proposition. If J is infinitesimally equivariant, then J : P → g∗ is a Poisson map. If J is generated by a left (respectively, right) action then we use the (+) (respectively, (−)) Lie–Poisson structure on g∗. The above development concerns momentum maps using the Hamilto- nian point of view. However, one can also consider them from the La- grangian point of view. In this context, we consider a Lie group G acting 2.7 Momentum Maps 41 on a configuration manifold Q and lift this action to the tangent bundle TQ using the tangent operation. Given a G-invariant Lagrangian L : TQ→ R, the corresponding momentum map is obtained by replacing the momentum pq in (2.7.1) with the fiber derivative FL(vq). Thus, J : TQ → g∗ is given by 〈J(vq), ξ〉 = 〈FL(vq), ξQ(q)〉 (2.7.4) or, in coordinates, Ja = ∂L ∂q˙i Kia, (2.7.5) where the action coefficients Kia are defined as before by writing ξQ(q i) = Kiaξ a∂/∂qi. 2.7.7 Proposition. For a solution of the Euler–Lagrange equations (even if the Lagrangian is degenerate), J is constant in time. Proof. In case L is a regular Lagrangian, this follows from its Hamilto- nian counterpart. It is useful to check it directly using Hamilton’s principle (which is the way it was originally done by Noether). To do this, choose any function φ(t, ) of two variables such that the conditions φ(a, ) = φ(b, ) = φ(t, 0) = 0 hold. Since L is G-invariant, for each Lie algebra element ξ ∈ g, the expression ∫ b a L(exp(φ(t, )ξ)q, exp(φ(t, )ξ)q˙)) dt (2.7.6) is independent of . Differentiating this expression with respect to at = 0 and setting φ′ = ∂φ/∂ taken at = 0, gives 0 = ∫ b a ( ∂L ∂qi ξiQφ ′ + ∂L ∂q˙i (TξQ · q˙)iφ′ ) dt. (2.7.7) Now we consider the variation q(t, ) = exp(φ(t, )ξ) · q(t). The correspond- ing infinitesimal variation is given by δq(t) = φ′(t)ξQ(q(t)). By Hamilton’s principle, we have 0 = ∫ b a ( ∂L ∂qi δqi + ∂L ∂q˙i ˙δqi ) dt. (2.7.8) Note that δ˙q = φ˙′ξQ + φ′(TξQ · q˙) and subtract (2.7.8) from (2.7.7) to give 0 = ∫ b a ∂L ∂q˙i (ξQ)iφ˙′ dt = − ∫ b a d dt ( ∂L ∂q˙i ξiQ ) φ′ dt. 42 2. A Crash Course in Geometric Mechanics Since φ′ is arbitrary, except for endpoint conditions, it follows that the integrand vanishes, and so the time derivative of the momentum map is zero and so the proposition is proved. 2.8 Symplectic and Poisson Reduction We have already seen how to use variational principles to reduce the Euler– Lagrange equations. On the Hamiltonian side, there are three levels of reduction of decreasing generality, that of Poisson reduction, symplectic reduction, and cotangent bundle reduction. Let us first consider Poisson reduction. For Poisson reduction we start with a Poisson manifold P and let the Lie group G act on P by Poisson maps. Assuming P/G is a smooth manifold, endow it with the unique Poisson structure on such that the canonical projection pi : P → P/G is a Poisson map. We can specify the Poisson structure on P/G explicitly as follows. For f and k : P/G→ R, let F = f ◦pi and K = k◦pi, so F and K are f and k thought of as G-invariant functions on P . Then {f, k}P/G is defined by {f, k}P/G ◦ pi = {F,K}P . (2.8.1) To show that {f, k}P/G is well defined, one has to prove that {F,K}P is G-invariant. This follows from the fact that F and K are G-invariant and the group action of G on P consists of Poisson maps. For P = T ∗G we get a very important special case. 2.8.1 Theorem (Lie–Poisson Reduction). Let P = T ∗G and assume that G acts on P by the cotangent lift of left translations. If one endows g∗ with the minus Lie–Poisson bracket, then P/G ∼= g∗. For symplectic reduction we begin with a symplectic manifold (P,Ω). Let G be a Lie group acting by symplectic maps on P ; in this case the action is called a symplectic action . Let J be an equivariant momentum map for this action and H a G-invariant Hamiltonian on P . Let Gµ = { g ∈ G | g · µ = µ } be the isotropy subgroup (symmetry subgroup) at µ ∈ g∗. As a consequence of equivariance, Gµ leaves J−1(µ) invariant. Assume for simplicity that µ is a regular value of J, so that J−1(µ) is a smooth manifold (see §2.8 below) and that Gµ acts freely and properly on J−1(µ), so that J−1(µ)/Gµ =: Pµ is a smooth manifold. Let iµ : J−1(µ) → P denote the inclusion map and let piµ : J−1(µ)→ Pµ denote the projection. Note that dimPµ = dimP − dimG− dimGµ. (2.8.2) Building on classical work of Jacobi, Liouville, Arnold and Smale, we have the following basic result of Marsden and Weinstein [1974] (see also Meyer [1973]). 2.8 Symplectic and Poisson Reduction 43 2.8.2 Theorem (Reduction Theorem). There is a unique symplectic structure Ωµ on Pµ satisfying i∗µΩ = pi ∗ µΩµ. (2.8.3) Given a G-invariant Hamiltonian H on P , define the reduced Hamilto- nian Hµ : Pµ → R by H = Hµ ◦ piµ. Then the trajectories of XH project to those of XHµ . An important problem is how to reconstruct trajectories of XH from trajectories of XHµ . Schematically, we have the situation in Figure 2.8.1. reduction reconstruction ? ? 6 6 P Pµ J−1(µ) iµ piµ Figure 2.8.1. Reduction to Pµ and reconstruction back to P . As we shall see later, the reconstruction process is where the holonomy and “geometric phase” ideas enter. In fact, we shall put a connection on the bundle piµ : J−1(µ)→ Pµ and it is through this process that one encounters the gauge theory point of view of mechanics. Let Oµ denote the coadjoint orbit through µ. As a special case of the symplectic reduction theorem, we get 2.8.3 Corollary. (T ∗G)µ ∼= Oµ. The symplectic structure inherited on Oµ is called the (Lie–Kostant– Kirillov) orbit symplectic structure . This structure is compatible with the Lie–Poisson structure on g∗ in the sense that the bracket of two func- tions on Oµ equals that obtained by extending them arbitrarily to g∗, taking the Lie–Poisson bracket on g∗ and then restricting to Oµ. Examples A. G = SO(3), g∗ = so(3)∗ ∼= R3. In this case the coadjoint action is the usual action of SO(3) on R3. This is because of the orthogonality of the elements of G. The set of orbits consists of spheres and a single point. The reduction process confirms that all orbits are symplectic manifolds. One calculates that the symplectic structure on the spheres is a multiple of the area element. 44 2. A Crash Course in Geometric Mechanics B. Jacobi–Liouville theorem. Let G = Tk be the k-torus and assume G acts on a symplectic manifold P . In this case the components of J are in involution and dimPµ = dimP − 2k, so 2k variables are eliminated. As we shall see, reconstruction allows one to reassemble the solution trajectories on P by quadratures in this Abelian case. C. Jacobi–Deprit elimination of the node. Let G = SO(3) act on P . In the classical case of Jacobi, P = T ∗R3 and in the generalization of Deprit [1983] one considers the phase space of n particles in R3. We just point out here that the reduced space Pµ has dimension dimP − 3 − 1 = dimP − 4 since Gµ = S1 (if µ 6= 0) in this case. The orbit reduction theorem of Marle [1976] and Kazhdan, Kostant, and Sternberg [1978] states that Pµ may be alternatively constructed as PO = J−1(O)/G, (2.8.4) where O ⊂ g∗ is the coadjoint orbit through µ. As above we assume we are away from singular points (see §2.8 below). The spaces Pµ and PO are isomorphic by using the inclusion map lµ : J−1(µ) → J−1(O) and taking equivalence classes to induce a symplectic isomorphism Lµ : Pµ → PO. The symplectic structure ΩO on PO is uniquely determined by j∗OΩ = pi ∗ OΩO + J ∗ OωO, (2.8.5) where jO : J−1(O)→ P is the inclusion, piO : J−1(O)→ PO is the projec- tion, and where JO = J|J−1(O) : J−1(O) → O and ωO is the orbit sym- plectic form. In terms of the Poisson structure, J−1(O)/G has the bracket structure inherited from P/G; in fact, J−1(O)/G is a symplectic leaf in P/G. Thus, we get the picture in Figure 2.8.2. /Gµ /G /G ? ? ? J−1(µ) ⊂ J−1(O) ⊂ P Pµ ∼= PO ⊂ P/G Figure 2.8.2. Orbit reduction gives another realization of Pµ. Kirillov has shown that every Poisson manifold P is the union of sym- plectic leaves, although the preceding construction explicitly realizes these symplectic leaves in this case by the reduction construction. A special case 2.9 Singularities and Symmetry 45 is the foliation of the dual g∗ of any Lie algebra g into its symplectic leaves, namely the coadjoint orbits. For example SO(3) is the union of spheres plus the origin, each of which is a symplectic manifold. Notice that the drop in dimension from T ∗ SO(3) to O is from 6 to 2, a drop of 4, as in general SO(3) reduction. An exception is the singular point, the origin, where the drop in dimension is larger. We turn to these singular points next. 2.9 Singularities and Symmetry 2.9.1 Proposition. Let (P,Ω) be a symplectic manifold, let G act on P by Poisson mappings, and let J : P → g∗ be a momentum map for this action (J need not be equivariant). Let Gz denote the symmetry group of z ∈ P defined by Gz = { g ∈ G | gz = z } and let gz be its Lie algebra, so gz = { ζ ∈ g | ζP (z) = 0 }. Then z is a regular value of J if and only if gz is trivial; that is, gz = {0}, or, equivalently, Gz is discrete. Proof. The point z is regular when the range of the linear map DJ(z) is all of g∗. However, ζ ∈ g is orthogonal to the range (in the sense of the g, g∗ pairing) if and only if for all v ∈ TzP , 〈ζ,DJ(z) · v〉 = 0 that is, d〈J, ζ〉(z) · v = 0 or Ω(X〈J,ζ〉(z), v) = 0 or Ω(ζP (z), v) = 0. As Ω is nondegenerate, ζ is orthogonal to the range iff ζP (z) = 0. The above proposition is due to Smale [1970]. It is the starting point of a large literature on singularities in the momentum map and singular re- duction. Arms, Marsden, and Moncrief [1981] show, under some reasonable hypotheses, that the level sets J−1(0) have quadratic singularities. As we shall see in the next chapter, there is a general shifting construction that enables one to effectively reduce J−1(µ) to the case J−1(0). In the finite dimensional case, this result can be deduced from the equivariant Darboux theorem, but in the infinite dimensional case, things are much more subtle. In fact, the infinite dimensional results were motivated by, and apply to, the singularities in the solution space of relativistic field theories such as gravity and the Yang–Mills equations (see Fischer, Marsden, and Moncrief [1980], Arms, Marsden, and Moncrief [1981, 1982] and Arms [1981]). The convexity theorem states that the image of the momentum map of a 46 2. A Crash Course in Geometric Mechanics torus action is a convex polyhedron in g∗; the boundary of the polyhedron is the image of the singular (symmetric) points in P ; the more symmetric the point, the more singular the boundary point. These results are due to Atiyah [1982] and Guillemin and Sternberg [1984] based on earlier con- vexity results of Kostant and the Shur–Horne theorem on eigenvalues of symmetric matrices. The literature on these topics and its relation to other areas of mathematics is vast. See, for example, Kirwan [1984a], Goldman and Millson [1990], Sjamaar [1990], Bloch, Flaschka, and Ratiu [1990], Sja- maar and Lerman [1991] and Lu and Ratiu [1991]. 2.10 A Particle in a Magnetic Field During cotangent bundle reduction considered in the next chapter, we shall have to add terms to the symplectic form called “magnetic terms”. To explain this terminology, we consider a particle in a magnetic field. Let B be a closed two-form on R3 and B = Bxi+Byj+Bzk the associated divergence free vector field, that is, iB(dx ∧ dy ∧ dz) = B, or B = Bxdy ∧ dz −Bydx ∧ dz +Bzdx ∧ dy. Thinking of B as a magnetic field, the equations of motion for a particle with charge e and mass m are given by the Lorentz force law : m dv dt = e c v ×B (2.10.1) where v = (x˙, y˙, z˙). On R3 × R3, that is, on (x,v)-space, consider the symplectic form ΩB = m(dx ∧ dx˙+ dy ∧ dy˙ + dz ∧ dz˙)− e c B. (2.10.2) For the Hamiltonian, take the kinetic energy: H = m 2 (x˙2 + y˙2 + z˙2) (2.10.3) writingXH(u, v, w) = (u, v, w, (u˙, v˙, w˙)), the condition definingXH , namely iXHΩB = dH is m(udx˙− u˙dx+ vdy˙ − v˙dy + wdz˙ − w˙dz) − e c [Bxvdz −Bxwdy −Byudz +Bywdx+Bzudy −Bzvdx] = m(x˙dx˙+ y˙dy˙ + z˙dz˙) (2.10.4) 2.10 A Particle in a Magnetic Field 47 which is equivalent to u = x˙, v = y˙, w = z˙, mu˙ = e(Bzv − Byw)/c, mv˙ = e(Bxw −Bzu)/c, and mw˙ = e(Byu−Bxv)/c, that is, to mx¨ = e c (Bz y˙ −By z˙) my¨ = e c (Bxz˙ −Bzx˙) (2.10.5) mz¨ = e c (Byx˙−Bxy˙) which is the same as (2.10.1). Thus the equations of motion for a particle in a magnetic field are Hamiltonian, with energy equal to the kinetic energy and with the symplectic form ΩB . If B = dA; that is, B = ∇ × A, where A is a one-form and A is the associated vector field, then the map (x,v) 7→ (x,p) where p = mv + eA/c pulls back the canonical form to ΩB , as is easily checked. Thus, equations (2.10.1) are also Hamiltonian relative to the canonical bracket on (x,p)-space with the Hamiltonian HA = 1 2m ‖p− e c A‖2. (2.10.6) Even in Euclidean space, not every magnetic field can be written as B = ∇ × A. For example, the field of a magnetic monopole of strength g 6= 0, namely B(r) = g r ‖r‖3 (2.10.7) cannot be written this way since the flux of B through the unit sphere is 4pig, yet Stokes’ theorem applied to the two hemispheres would give zero. Thus, one might think that the Hamiltonian formulation involving only B (i.e., using ΩB and H) is preferable. However, one can recover the magnetic potential A by regarding A as a connection on a nontrivial bundle over R3\{0}. The bundle over the sphere S2 is in fact the same Hopf fibration S3 → S2 that we encountered in §1.6. This same construction can be carried out using reduction. For a readable account of some aspects of this situation, see Yang [1980]. For an interesting example of Weinstein in which this monopole comes up, see Marsden [1981, p. 34]. When one studies the motion of a colored (rather than a charged) particle in a Yang–Mills field, one finds a beautiful generalization of this construc- tion and related ideas using the theory of principal bundles; see Sternberg [1977], Weinstein [1978b] and Montgomery [1984, 1986]. In the study of centrifugal and Coriolis forces one discovers some structures analogous to those here (see Marsden and Ratiu [1999] for more information). We shall return to particles in Yang–Mills fields in our discussion of optimal control in Chapter 7. 48 2. A Crash Course in Geometric Mechanics Page 49 3 Tangent and Cotangent Bundle Reduction In this chapter we discuss the cotangent bundle reduction theorem. Versions of this are already given in Smale [1970], but primarily for the Abelian case. This was amplified in the work of Satzer [1977] and motivated by this, was extended to the nonabelian case in Abraham and Marsden [1978]. An important formulation of this was given by Kummer [1981] in terms of connections. Building on this, the “bundle picture” was developed by Montgomery, Marsden, and Ratiu [1984], Montgomery [1986] and Cendra, Marsden, and Ratiu [2001]. From the symplectic viewpoint, the principal result is that the reduction of a cotangent bundle T ∗Q at µ ∈ g∗ is a bundle over T ∗(Q/G) with fiber the coadjoint orbit through µ. Here, S = Q/G is called shape space . From the Poisson viewpoint, this reads: (T ∗Q)/G is a g∗-bundle over T ∗(Q/G), or a Lie–Poisson bundle over the cotangent bundle of shape space. We de- scribe the geometry of this reduction using the mechanical connection and explicate the reduced symplectic structure and the reduced Hamiltonian for simple mechanical systems. On the Lagrangian, or tangent bundle, side there is a counterpart of both symplectic and Poisson reduction in which one concentrates on the variational principle rather than on the symplectic and Poisson structures. We will develop this as well. 50 3. Tangent and Cotangent Bundle Reduction 3.1 Mechanical G-systems By a symplectic (resp., Poisson) G-system we mean a symplectic (resp., Poisson) manifold (P,Ω) together with the symplectic action of a Lie group G on P , an equivariant momentum map J : P → g∗ and a G-invariant Hamiltonian H : P → R. Following terminology of Smale [1970], we refer to the following special case of a symplectic G-system as a simple mechanical G-system . We choose P = T ∗Q, assume there is a Riemannian metric 〈〈 , 〉〉 on Q, that G acts on Q by isometries (and so G acts on T ∗Q by cotangent lifts) and that H(q, p) = 1 2 ‖p‖2q + V (q), (3.1.1) where ‖ · ‖q is the norm induced on T ∗qQ, and where V is a G-invariant potential. We abuse notation slightly and write either z = (q, p) or z = pq for a covector based at q ∈ Q and we shall also let ‖ · ‖q denote the norm on TqQ. Points in TqQ shall be denoted vq or (q, v) and the pairing between T ∗qQ and TqQ is simply written 〈pq, vq〉, 〈p, v〉, or 〈(q, p), (q, v)〉. (3.1.2) Other natural pairings between spaces and their duals are also denoted 〈 , 〉. The precise meaning will always be clear from the context. Unless mentioned to the contrary, the default momentum map for simple mechanical G-systems will be the standard one: J : T ∗Q→ g∗, where 〈J(q, p), ξ〉 = 〈p, ξQ(q)〉 (3.1.3) and where ξQ denotes the infinitesimal generator of ξ on Q. For the convenience of the reader we will write most formulas in both coordinate free notation and in coordinates using standard tensorial con- ventions. As in Chapter 2, coordinate indices for P are denoted zI , zJ , etc., on Q by qi, qj , etc., and on g (relative to a vector space basis of g) by ξa, ξb, etc. For instance, z˙ = XH(z) can be written z˙I = XIH(z J) or z˙I = {zI , H}. (3.1.4) Recall from Chapter 2 that the Poisson tensor is defined by B(dH) = XH , or equivalently, BIJ(z) = {zI , zJ}, so (3.1.4) reads XIH = B IJ ∂H ∂zJ or z˙I = BIJ(z) ∂H ∂zJ (z). (3.1.5) Equation (3.1.1) reads H(q, p) = 1 2 gijpipj + V (q) (3.1.6) 3.1 Mechanical G-systems 51 and (3.1.2) reads 〈p, v〉 = pivi, (3.1.7) while (3.1.3) reads Ja(q, p) = piKia(q) (3.1.8) where the action tensor Kia is defined, as in Chapter 2, by [ξQ(q)]i = Kia(q)ξ a. (3.1.9) The Legendre transformation is denoted FL : TQ → T ∗Q and in the case of simple mechanical systems, is simply the metric tensor regarded as a map from vectors to covectors; in coordinates, FL(q, v) = (q, p), where pi = gijvj . (3.1.10) Examples A. The spherical pendulum. Here Q = S2, the sphere on which the bob moves, the metric is the standard one, the potential is the gravitational potential and G = S1 acts on S2 by rotations about the vertical axis. The momentum map is simply the angular momentum about the z-axis. We will work this out in more detail below. B. The double spherical pendulum. Here the configuration space is Q = S2 × S2, the group is S1 acting by simultaneous rotation about the z-axis and again the momentum map is the total angular momentum about the z-axis. Again, we will see more detail below. C. Coupled rigid bodies. Here we have two rigid bodies in R3 coupled by a ball in socket joint. We choose Q = R3 × SO(3) × SO(3) describing the joint position and the attitude of each of the rigid bodies relative to a reference configuration B. The Hamiltonian is the kinetic energy, which defines a metric on Q. Here G = SE(3) which acts on the left in the obvious way by transforming the positions of the particles xi = AiX + w, i = 1, 2 by Euclidean motions. The momentum map is the total linear and angular momentum. If the bodies have additional material symmetry, G is correspondingly enlarged; cf . Patrick [1989]. See Figure 3.1.1. D. Ideal fluids. For an ideal fluid moving in a container represented by a region Ω ⊂ R3, the configuration space is Q = Diffvol(Ω), the volume pre- serving diffeomorphisms of Ω to itself. Here G = Q acts on itself on the right and H is the total kinetic energy of the fluid. Here Lie–Poisson reduction is relevant. We refer to Ebin and Marsden [1970] for the relevant functional analytic technicalities. We also refer to Marsden and Weinstein [1982, 1983] 52 3. Tangent and Cotangent Bundle Reduction z x y w B X 1 2 Figure 3.1.1. The configuration space for the dynamics of two coupled rigid bodies. for more information and corresponding ideas for plasma physics, and to Marsden and Hughes [1983] and Simo, Marsden, and Krishnaprasad [1988] for elasticity. The examples we will focus on in these lectures are the spherical pendula and the classical water molecule. Let us pause to give some details for the later example. 3.2 The Classical Water Molecule We started discussing this example in Chapter 1. The “primitive” configu- ration space of this system is Q = R3 × R3 × R3, whose points are triples, denoted (r, r1, r2), giving the positions of the three masses relative to an inertial frame. For simplicity, we ignore any singular behavior that may happen during collisions, as this aspect of the system is not of concern to the present discussion. The Lagrangian is given on TQ by L(r, r1, r2, r˙, r˙1, r˙2) = 1 2 {M‖r˙‖2 +m‖r˙1‖2 +m‖r˙2‖2}−V (r, r1, r2) (3.2.1) where V is a potential. 3.2 The Classical Water Molecule 53 The Legendre transformation produces a Hamiltonian system on Z = T ∗Q with H(r, r1, r2,P,p1,p2) = ‖P‖2 2M + ‖p1‖2 2m + ‖p2‖2 2m + V. (3.2.2) The Euclidean group acts on this system by simultaneous translation and rotation of the three component positions and momenta. In addition, there is a discrete symmetry closely related to interchanging r1 and r2 (and, si- multaneously, p1 and p2). Accordingly, we will assume that V is symmetric in its second two arguments as well as being Euclidean-invariant. The translation group R3 acts on Z by a · (r, r1, r2,P,p1,p2) = (r + a, r1 + a, r2 + a,P,p1,p2) and the corresponding momentum map J : Z → R3 is the total linear momentum J(r, r1, r2,P,p1,p2) = P + p1 + p2. (3.2.3) Reduction to center of mass coordinates entails that we set J equal to a constant and quotient by translations. To coordinatize the reduced space (and bring relevant metric tensors into diagonal form), we shall use Jacobi– Bertrand–Haretu coordinates, namely r = r2 − r1 and s = r− 12(r1 + r2) (3.2.4) as in Figure 3.2.1. m M m Figure 3.2.1. Jacobi-Bertrand-Haretu coordinates. To determine the correct momenta conjugate to these variables, we go back to the Lagrangian and write it in the variables r, s, r˙, s˙, using the relations r˙ = r˙2 − r˙1, s˙ = r˙− 12(r˙1 + r˙2) 54 3. Tangent and Cotangent Bundle Reduction and J = m(r˙1 + r˙2) +M r˙, which give r˙1 = r˙− s˙− 12 r˙ r˙2 = r˙− s˙ + 12 r˙ r˙ = 1 M (J + 2ms˙) where M = M + 2m is the total mass. Substituting into (3.2.1) and sim- plifying gives L = ‖J‖2 2M + m 4 ‖r˙‖2 +Mm‖s˙‖2 − V (3.2.5) where m = m/M and where one expresses the potential V in terms of r and s (if rotationally invariant, it depends on ‖r‖, ‖s‖, and r · s alone). A crucial feature of the above choice is that the Riemannian metric (mass matrix ) associated to L is diagonal as (3.2.5) shows. The conjugate momenta are pi = ∂L ∂r˙ = m 2 r˙ = 1 2 (p2 − p1) (3.2.6) and, with M = M/M, σ = ∂L ∂s˙ = 2Mms˙ = 2mP−M(p1 + p2) = P−MJ = 2mJ− p1 − p2. (3.2.7) We next check that this is consistent with Hamiltonian reduction. First, note that the symplectic form before reduction, Ω = dr1 ∧ dp1 + dr2 ∧ dp2 + dr ∧ dP can be written, using the relation p1 + p2 + P = J, as Ω = dr ∧ dpi + ds ∧ dσ + dr¯ ∧ dJ (3.2.8) where r¯ = 12 (r1 + r2) +Ms is the system center of mass. Thus, if J is a constant (or we pull Ω back to the surface J = constant), we get the canonical form in (r, pi) and (s, σ) as pairs of conjugate variables, as expected. Second, we compute the reduced Hamiltonian by substituting into H = 1 2m ‖p1‖2 + 12m‖p2‖ 2 + 1 2M ‖P‖2 + V 3.2 The Classical Water Molecule 55 the relations p1 + p2 + P = J, pi = 1 2 (p2 − p1), and σ = P−MJ. One obtains H = 1 4mM ‖σ‖2 + ‖pi‖ 2 m + 1 2M‖J‖ 2 + V. (3.2.9) Notice that in this case the reduced Hamiltonian and Lagrangian are Leg- endre transformations of one another . Remark. There are other choices of canonical coordinates for the system after reduction by translations, but they might not be as convenient as the above Jacobi–Bertrand–Haretu coordinates. For example, if one uses the apparently more “democratic” choice s1 = r1 − r and s2 = r2 − r, then their conjugate momenta are pi1 = p1 −mJ and pi2 = p2 −mJ. One finds that the reduced Lagrangian is L = 1 2 m′′‖s˙1‖2 + 12m ′′‖s˙2‖2 −mms˙1 · s˙2 − V and the reduced Hamiltonian is H = 1 2m′ ‖pi1‖2 + 12m′ ‖pi2‖ 2 + 1 M pi1 · pi2 + V where m′ = mM/(m + M) and m′′ = mMm/m′. It is the cross terms in H and L that make these expressions inconvenient. Next we turn to the action of the rotation group, SO(3) and the role of the discrete group that “swaps” the two masses m. Our phase space is Z = T ∗(R3 × R3) parametrized by (r, s, pi, σ) with the canonical symplectic structure. The group G = SO(3) acts on Z by the cotangent lift of rotations on R3 ×R3. The corresponding infinitesimal action on Q is ξQ(r, s) = (ξ × r, ξ × s) and the corresponding momentum map is J : Z → R3, where 〈J(r, s, pi, σ), ξ〉 = pi · (ξ × r) + σ · (ξ × s) = ξ · (r× pi + s× σ). 56 3. Tangent and Cotangent Bundle Reduction In terms of the original variables, we find that J = r× pi + s× σ = (r2 − r1)× 12(p2 − p1) + ( r− 1 2 (r1 + r2) ) × (P−MJ) (3.2.10) which becomes, after simplification, J = r1 × (p1 −mJ) + r2 × (p2 −mJ) + r× (P−MJ) (3.2.11) which is the correct total angular momentum of the system. Remark. This expression for J also equals s1 × pi1 + s2 × pi2 in terms of the alternative variables discussed above. 3.3 The Mechanical Connection In this section we work in the context of a simple mechanical G-system. We shall also assume that G acts freely on Q so we can regard Q→ Q/G as a principal G-bundle. (We make some remarks on this assumption later.) For each q ∈ Q, let the locked inertia tensor be the map I(q) : g→ g∗ defined by 〈I(q)η, ζ〉 = 〈〈ηQ(q), ζQ(q)〉〉. (3.3.1) Since the action is free, I(q) is an inner product. The terminology comes from the fact that for coupled rigid or elastic systems, I(q) is the classical moment of inertia tensor of the rigid body obtained by locking all the joints of the system. In coordinates, Iab = gijKiaK j b. (3.3.2) Define the map A : TQ→ g by assigning to each (q, v) the corresponding angular velocity of the locked system : A(q, v) = I(q)−1(J(FL(q, v))). (3.3.3) In coordinates, Aa = IabgijKibvj . (3.3.4) One can think of Aq (the restriction of A to TqQ) as the map induced by the momentum map via two Legendre transformations, one on the given system and one on the instantaneous rigid body associated with it, as in Figure 3.3.1. 3.3.1 Proposition. The map A is a connection on the principal G-bundle Q→ Q/G. 3.3 The Mechanical Connection 57 T ∗Q g∗ TQ g J Aq FLq I(q) - - 6 6 Figure 3.3.1. The diagram defining the mechanical connection. In other words, A is G-equivariant and satisfies A(ξQ(q)) = ξ, both of which are readily verified. A concise and convenient summary of principal connections is given in Bloch [2003]. In checking equivariance one uses invariance of the metric; that is, equiv- ariance of FL : TQ→ T ∗Q, equivariance of J : T ∗Q→ g∗, and equivariance of I in the sense of a map I : Q→ L(g, g∗), namely I(gq) ·Adgξ = Ad∗g−1I(q) · ξ. We call A the mechanical connection . This connection is implicitly used in the work of Smale [1970], and Abraham and Marsden [1978], and ex- plicitly in Kummer [1981], Guichardet [1984], Shapere and Wilczeck [1989], Simo, Lewis, and Marsden [1991], and Montgomery [1990]. We note that all of the preceding formulas are given in Abraham and Marsden [1978, Section 4.5], but not from the point of view of connections. The horizontal space horq of the connection A at q ∈ Q is given by the kernel of Aq; thus, by (3.3.3), horq = { (q, v) | J(FL(q, v)) = 0 }. (3.3.5) Using the formula (3.1.3) for J, we see that horq is the space orthogonal to the G-orbits, or, equivalently, the space of states with zero total angular momentum in the case G = SO(3) (see Figure 3.3.2). This may be used to define the mechanical connection—in fact, principal connections are char- acterized by their horizontal spaces. The vertical space consists of vectors that are mapped to zero under the projection Q→ S = Q/G; that is, verq = { ξQ(q) | ξ ∈ g }. (3.3.6) For each µ ∈ g∗, define the 1-form Aµ on Q by 〈Aµ(q), v〉 = 〈µ,A(q, v)〉 (3.3.7) that is, (Aµ)i = gijKjbµaIab. (3.3.8) This coordinate formula shows that Aµ is given by (I−1µ)Q. We prove this intrinsically in the next Proposition. 58 3. Tangent and Cotangent Bundle Reduction q Q G .q Q /G[q] Figure 3.3.2. The horizontal space of the mechanical connection. 3.3.2 Proposition. The one form Aµ takes values in J−1(µ). Moreover, identifying vectors and one-forms, Aµ = (I−1µ)Q. Proof. The first part holds for any connection on Q→ Q/G and follows from the property A(ξQ)=ξ. Indeed, 〈J(Aµ(q)), ξ〉 = 〈Aµ(q), ξQ(q)〉 = 〈µ,A(ξQ(q))〉 = 〈µ, ξ〉. The second part follows from the definitions: 〈Aµ(g), v〉 = 〈µ,A(v)〉 = 〈µ, I−1(q)J(FL(v))〉 = 〈I−1(q)µ,J(FL(v))〉 = 〈(I−1(q)µ)Q(q),FL(v)〉 = 〈〈(I−1(q)µ)Q(q), v〉〉q. Notice that this proposition holds for any connection on Q → Q/G, not just the mechanical connection. It is straightforward to see that Aµ is characterized by K(Aµ(q)) = inf{K(q, β) | β ∈ J−1q (µ) } (3.3.9) where Jq = J|T ∗qQ and K(q, p) = 12‖p‖2q is the kinetic energy function (see Figure 3.3.3). Indeed (3.3.9) means that Aµ(q) lies in and is orthogonal to 3.3 The Mechanical Connection 59 the affine space J−1q (µ), that is, is orthogonal to the linear space Jq(µ) − Aµ(q) = J−1q (0). The latter means Aµ(q) is in the vertical space, so is of the form ηq and the former means η is given by I−1(q)µ. Q q Figure 3.3.3. The mechanical connection as the orthogonal vector to the level set of J in the cotangent fiber. The horizontal-vertical decomposition of a vector (q, v) ∈ TqQ is given by the general prescription v = horq v + verqv (3.3.10) where verqv = [A(q, v)]Q(q) and horq v = v − verqv. In terms of T ∗Q rather than TQ, we define a map ω : T ∗Q→ g by ω(q, p) = I(q)−1J(q, p) (3.3.11) that is, ωa = IabKibpi, (3.3.12) and, using a slight abuse of notation, a projection hor : T ∗Q→ J−1(0) by hor(q, p) = p−AJ(q,p)(q) (3.3.13) that is, (hor(q, p))i = pi − gijKjbpkKkaIab. (3.3.14) This map hor in (3.3.13) will play a fundamental role in what follows. We also refer to hor as the shifting map. This map will be used in the proof of the cotangent bundle reduction theorem in §3.4; it is also an essential ingredient in the description of a particle in a Yang–Mills field via the Kaluza–Klein construction, generalizing the electromagnetic case in which p 7→ p− ecA. This aspect will be brought in later. 60 3. Tangent and Cotangent Bundle Reduction Example. Let Q = G be a Lie group with symmetry group G itself, acting say on the left. In this case, Aµ is independent of the Hamiltonian. Since Aµ(q) ∈ J−1(µ), it follows that Aµ is the right invariant one form whose value at the identity e is µ. (This same one form Aµ was used in Marsden and Weinstein [1974].) The curvature curv A of the connection A is the covariant exterior derivative of A, defined to be the exterior derivative acting on the hor- izontal components. The curvature may be regarded as a measure of the lack of integrability of the horizontal subbundle. At q ∈ Q, (curvA)(v, w) = dA(hor v,hor w) = −A([hor v,hor w]), (3.3.15) where the Jacobi–Lie bracket is computed using extensions of v, w to vector fields. The first equality in (3.3.9) is the definition and the second follows from the general formula dγ(X,Y ) = X[γ(Y )]− γ([X,Y ]) relating the exterior derivative of a one form and Lie brackets. Taking the µ-component of (3.3.15), we get a 2-form 〈µ, curvA〉 on Q given by 〈µ, curvA〉(v, w) = −Aµ([hor v,hor w]) = −Aµ([v −A(v)Q, w −A(w)Q]) = Aµ([v,A(w)Q]−Aµ([w,A(v)Q] −Aµ([v, w])− 〈µ, [A(v),A(w)]〉. (3.3.16) In (3.3.16) we may choose v and w to be extended by G-invariance. Then A(w)Q = ζQ for a fixed ζ, so [v,A(w)Q] = 0, so we can replace this term by v[Aµ(w)], which is also zero, noting that v[〈µ, ξ〉] = 0, as 〈µ, ξ〉 is constant. Thus, (3.3.16) gives the Cartan structure equation : 〈µ, curvA〉 = dAµ − [A,A]µ (3.3.17) where [A,A]µ(v, w) = 〈µ, [A(v),A(w)]〉, the bracket being the Lie algebra bracket. The amended potential Vµ is defined by Vµ = H ◦ Aµ; (3.3.18) this function also plays a crucial role in what follows. In coordinates, Vµ(q) = V (q) + 1 2 Iab(q)µaµb (3.3.19) or, intrinsically Vµ(q) = V (q) + 1 2 〈µ, I(q)−1µ〉. (3.3.20) 3.4 The Geometry and Dynamics of Cotangent Bundle Reduction 61 3.4 The Geometry and Dynamics of Cotangent Bundle Reduction Given a symplectic G-system (P,Ω, G,J, H), in Chapter 2 we defined the reduced space Pµ to be Pµ = J−1(µ)/Gµ, (3.4.1) assuming µ is a regular value of J (or a weakly regular value) and that the Gµ action is free and proper (or an appropriate slice theorem applies) so that the quotient (3.4.1) is a manifold. The reduction theorem states that the symplectic structure Ω on P naturally induces one on Pµ; it is denoted Ωµ. Recall that Pµ ∼= PO where PO = J−1(O)/G and O ⊂ g∗ is the coadjoint orbit through µ. Next, map J−1(O) → J−1(0) by the map hor given in the last section. This induces a map on the quotient spaces by equivariance; we denote it horO: horO : J−1(O)/G→ J−1(0)/G. (3.4.2) Reduction at zero is simple: J−1(0)/G is isomorphic with T ∗(Q/G) by the following identification: βq ∈ J−1(0) satisfies 〈βq, ξQ(q)〉 = 0 for all ξ ∈ g, so we can regard βq as a one form on T (Q/G). As a set, the fiber of the map horO is identified with O. Therefore, we have realized (T ∗Q)O as a coadjoint orbit bundle over T ∗(Q/G). The spaces are summarized in Figure 3.4.1. /Gµ /G piO ? ? ? ? (T ∗Q)µ ∼= (T ∗Q)O injection horµ horO surjection ? ? ? T ∗(Q/Gµ) T ∗(Q/G) ? T ∗Q ⊂ J−1(µ) ⊂ J−1(O) ⊂ T ∗Q Q −→ Q/Gµ −→ Q/G ←− Q Figure 3.4.1. Cotangent bundle reduction. The Poisson bracket structure of the bundle horO as a synthesis of the Lie–Poisson structure, the cotangent structure, the magnetic and inter- action terms is investigated in Montgomery, Marsden, and Ratiu [1984], Montgomery [1986] and Cendra, Marsden, and Ratiu [2001]. 62 3. Tangent and Cotangent Bundle Reduction For the symplectic structure, it is a little easier to use Pµ. Here, we restrict the map hor to J−1(µ) and quotient by Gµ to get a map of Pµ to J−1(0)/Gµ. If Jµ denotes the momentum map for Gµ, then J−1(0)/Gµ embeds in J−1µ (0)/Gµ ∼= T ∗(Q/Gµ). The resulting map horµ embeds Pµ into T ∗(Q/Gµ). This map is induced by the shifting map: pq 7→ pq −Aµ(q). (3.4.3) The symplectic form on Pµ is obtained by restricting the form on T ∗(Q/Gµ) given by Ωcanonical − dAµ. (3.4.4) The two form dAµ drops to a two form βµ on the quotient, so (3.4.4) defines the symplectic structure of Pµ. We call βµ the magnetic term following the ideas from §2.10 and the terminology of Abraham and Marsden [1978]. We also note that on J−1(µ) (and identifying vectors and covectors via FL), [A(v),A(w)] = [µ, µ] = 0, so that the form βµ may also be regarded as the form induced by the µ-component of the curvature. In the PO context, the distinction between dAµ and (curv α)µ becomes important. We shall see this aspect in Chapter 4. Two limiting cases. The first (that one can associate with Arnold [1966]) is when Q = G in which case PO ∼= O and the base is trivial in the PO → T ∗(Q/G) picture, while in the Pµ → T ∗(Q/Gµ) picture, the fiber is trivial and the base space is Q/Gµ ∼= O. Here the description of the orbit symplectic structure induced by dαµ coincides with that given by Kirillov [1976]. The other limiting case (that one can associate with Smale [1970]) is when G = Gµ; for instance, this holds in the Abelian case. Then Pµ = PO = T ∗(Q/G) with symplectic form Ωcanonical − βµ.1 Note that the construction of Pµ requires only that the Gµ action be free; then one gets Pµ embedded in T ∗(Q/Gµ). However, the bundle picture of PO → T ∗(Q/G) with fiber O requires G to act freely on Q, so Q/G is defined. If the G-action is not free, one can either deal with singularities or use Montgomery’s method above, replacing shape space Q/G with the 1 Richard Montgomery has given an interesting condition, generalizing this Abelian case and allowing non-free actions, which guarantees that Pµ is still a cotangent bundle. This condition (besides technical assumptions guaranteeing the objects in question are manifolds) is dim g − dim gµ = 2(dim gQ − dim gµQ), where gQ is the isotropy algebra of the G-action on Q (assuming gQ has constant dimension) and g µ Q is that for Gµ. The result is that Pµ is identified with T ∗(Qµ/Gµ), where Qµ is the projection of J−1(µ) ⊂ T ∗Q to Q under the cotangent bundle projection. (For free actions of G on Q, Qµ = Q.) The proof is a modification of the one given above. 3.4 The Geometry and Dynamics of Cotangent Bundle Reduction 63 modified shape space Qµ/G. For example, already in the Kepler problem in which G = SO(3) acts on Q = R3\{0}, the action of G on Q is not free; for µ 6= 0, Qµ is the plane orthogonal to µ (minus {0}). On the other hand, Gµ = S1, the rotations about the axis µ 6= 0, acts freely on J−1(µ), so Pµ is defined. But Gµ does not act freely on Q! Instead, Pµ is actually T ∗(Qµ/Gµ) = T ∗(0,∞), with the canonical cotangent structure. The case µ = 0 is singular and is more interesting! (See Marsden [1981].) Semidirect Products. Another context in which one can get interesting reduction results, both mathematically and physically, is that of semi-direct products ( Marsden, Ratiu, and Weinstein [1984a, 1984b]). Here we consider a linear representation of G on a vector space V . We form the semi-direct product S = GsV and look at its coadjoint orbit Oµ,a through (µ, a) ∈ g∗ × V ∗. The semi-direct product reduction theorem says that Oµ,a is symplectomorphic with the reduced space obtained by reducing T ∗G at µa = µ|ga by the sub-group Ga (the isotropy for the action of G on V ∗ at a ∈ V ∗). If Ga is Abelian (the generic case) then Abelian reduction gives Oµ,a ∼= T ∗(G/Ga) with the canonical plus magnetic structure. For example, the generic orbits in SE(3) = SO(3)sR3 are cotangent bundles of spheres; but the orbit symplectic structure has a non-trivial magnetic term—see Marsden and Ratiu [1999] for their computation. Semi-direct products come up in a variety of interesting physical sit- uations, such as the heavy rigid body, compressible fluids and MHD; we refer to Marsden, Ratiu, and Weinstein [1984a, 1984b] and Holm, Marsden, Ratiu, and Weinstein [1985] for details. For semidirect products, one has a useful result called reduction by stages. Namely, one can reduce by S in two successive stages, first by V and then by G. For example, for SE(3) one can reduce first by translations, and then by rotations, and the result is the same as reducing by SE(3). For a general reduction by stages result, see Marsden, Misiolek, Perlmutter, and Ratiu [1998]. For a Lagrangian analogue of semidirect product theory, see Holm, Mars- den, and Ratiu [1998b] and for Lagrangian reduction by stages, see Cendra, Marsden, and Ratiu [2001]. The Dynamics of Cotangent Bundle Reduction. Given a symplec- tic G-system, we get a reduced Hamiltonian system on Pµ ∼= PO obtained by restricting H to J−1(µ) or J−1(O) and then passing to the quotient. This produces the reduced Hamiltonian function Hµ and thereby a Hamil- tonian system on Pµ. The resulting vector field is the one obtained by restricting and projecting the Hamiltonian vector field XH on P . The re- sulting dynamical system XHµ on Pµ is called the reduced Hamiltonian system . 64 3. Tangent and Cotangent Bundle Reduction Let us compute Hµ in each of the pictures Pµ and PO. In either case the shift by the map hor is basic, so let us first compute the function on J−1(0) given by HAµ(q, p) = H(q, p+Aµ(q)). (3.4.5) Indeed, HAµ(q, p) = 1 2 〈〈p+Aµ, p+Aµ〉〉q + V (q) = 1 2 ‖p‖2q + 〈〈p,Aµ〉〉q + 1 2 ‖Aµ‖2q + V (q). (3.4.6) If p = FL · v, then 〈〈p,Aµ〉〉q = 〈Aµ, v〉 = 〈µ,A(q, v)〉 = 〈µ, I(q)J(p)〉 = 0 since J(p) = 0. Thus, on J−1(0), HAµ(q, p) = 1 2 ‖p‖2q + Vµ(q). (3.4.7) In T ∗(Q/Gµ), we obtain Hµ by selecting any representative (q, p) of T ∗(Q/Gµ) in J−1(0) ⊂ T ∗Q, shifting it to J−1(µ) by p 7→ p + Aµ(q) and then calculating H. Thus, the above calculation (3.4.7) proves: 3.4.1 Proposition. The reduced Hamiltonian Hµ is the function ob- tained by restricting to the affine symplectic subbundle Pµ ⊂ T ∗(Q/Gµ), the function Hµ(q, p) = 1 2 ‖p‖2 + Vµ(q) (3.4.8) defined on T ∗(Q/Gµ) with the symplectic structure Ωµ = Ωcan − βµ (3.4.9) where βµ is the two form on Q/Gµ obtained from dAµ on Q by passing to the quotient. Here we use the quotient metric on Q/Gµ and identify Vµ with a function on Q/Gµ. Example. If Q = G and the symmetry group is G itself, then the reduc- tion construction embeds Pµ ⊂ T ∗(Q/Gµ) as the zero section. In fact, Pµ is identified with Q/Gµ ∼= G/Gµ ∼= Oµ. To describe Hµ on J−1(O)/G is easy abstractly; one just calculates H restricted to J−1(O) and passes to the quotient. More concretely, we choose an element [(q, p)] ∈ T ∗(Q/G), where we identify the representative with an element of J−1(0). We also choose an element ν ∈ O, a coadjoint orbit, and shift (q, p) 7→ (q, p+Aν(q)) to a point in J−1(O). Thus, we get: 3.4.2 Proposition. Regarding Pµ ∼= PO as an O-bundle over T ∗(Q/G), the reduced Hamiltonian is given by HO(q, p, ν) = 1 2 ‖p‖2 + Vν(q) 3.5 Examples 65 where (q, p) is a representative in J−1(0) of a point in T ∗(Q/G) and where ν ∈ O. The symplectic structure in this second picture was described abstractly above. To describe it concretely in terms of T ∗(Q/G) and O is more diffi- cult; this problem was solved by Montgomery, Marsden, and Ratiu [1984]. See also Cendra, Marsden, and Ratiu [2001], Marsden and Perlmutter [1999] and Zaalani [1999] for related issues. For a Lagrangian version of this theo- rem, see below and Marsden, Ratiu and Scheurle [2000]. We shall see some special aspects of this structure when we separate internal and rotational modes in the next chapters. 3.5 Examples We next give some basic examples of the cotangent bundle reduction con- struction. The examples will be the spherical pendulum, the double spher- ical pendulum, and the water molecule. Of course many more examples can be given and we refer the reader to the literature cited for a wealth of information. The ones we have chosen are, we hope, simple enough to be of pedagogical value, yet interesting. A. The Spherical Pendulum. Here, Q = S2 with the standard metric as a sphere of radius R in R3, V is the gravitational potential for a mass m, and G = S1 acts on Q by rotations about the vertical axis. To make the action free, one has to delete the north and south poles. Relative to coordinates θ, ϕ as in Figure 3.5.1, we have V (θ, ϕ) = −mgR cos θ. We claim that the mechanical connection A : TQ→ R is given by A(θ, ϕ, θ˙, ϕ˙) = ϕ˙. (3.5.1) To see this, note that ξQ(θ, ϕ) = (θ, ϕ, 0, ξ) (3.5.2) since G = S1 acts by rotations about the z-axis: (θ, ϕ) 7→ (θ, ϕ + ψ). The metric is 〈〈(θ, ϕ, θ˙1, ϕ˙1), (θ, ϕ, θ˙2, ϕ˙2)〉〉 = mR2θ˙1θ˙2 +mR2 sin2 θϕ˙1ϕ˙2, (3.5.3) which is m times the standard inner product of the corresponding vectors in R3. The momentum map is the angular momentum about the z-axis: J : T ∗Q→ R; J(θ, ϕ, pθ, pϕ) = pϕ. (3.5.4) The Legendre transformation is pθ = mR2θ˙, pϕ = (mR2 sin2 θ)ϕ˙ (3.5.5) 66 3. Tangent and Cotangent Bundle Reduction x z y R Figure 3.5.1. The configuration space of the spherical pendulum is the two sphere. and the locked inertia tensor is 〈I(θ, ϕ)η, ζ〉 = 〈〈(θ, ϕ, 0, η)(θ, ϕ, 0, ζ)〉〉 = (mR2 sin2 θ)ηζ. (3.5.6) Note that I(θ, ϕ) = m(R sin θ)2 is the (instantaneous) moment of inertia of the mass m about the z-axis. In this example, we identify Q/S1 with the interval [0, pi]; that is, the θ-variable. In fact, it is convenient to regard Q/S1 as S1 mod Z2, where Z2 acts by reflection θ 7→ −θ. This helps to understand the singularity in the quotient space. For µ ∈ g∗ ∼= R, the one form Aµ is given by (3.3.3) and (3.3.7) as Aµ(θ, ϕ) = µdϕ. (3.5.7) From (3.5.4), one sees directly that Aµ takes values in J−1(µ). The shifting map is given by (3.3.13): hor(θ, ϕ, pθ, pϕ) = (θ, ϕ, pθ, 0). The curvature of the connection A is zero in this example because shape space is one dimensional. The amended potential is Vµ(θ) = V (θ, ϕ) + 1 2 〈µ, I(θ, ϕ)−1µ〉 = −mgR cos θ + 1 2 µ2 mR2 sin2 θ and so the reduced Hamiltonian on T ∗(S1/Z2) is Hµ(θ, pθ) = 1 2 p2θ mR2 + Vµ(θ). (3.5.8) 3.5 Examples 67 The reduced Hamiltonian equations are therefore θ˙ = pθ mR2 , and p˙θ = −mgR sin θ + cos θ sin3 θ µ2 mR2 . (3.5.9) Note that the extra term is singular at θ = 0. B. The Double Spherical Pendulum. Consider the mechanical sys- tem consisting of two coupled spherical pendula in a gravitational field (see Figure 3.5.2). x z y gravity Figure 3.5.2. The configuration space for the double spherical pendulum consists of two copies of the two sphere. Let the position vectors of the individual pendula relative to their joints be denoted q1 and q2 with fixed lengths l1 and l2 and with masses m1 and m2. The configuration space is Q = S2l1 × S2l2 , the product of spheres of radii l1 and l2 respectively. Since the “absolute” vector for the second pendulum is q1 + q2, the Lagrangian is L(q1,q2, q˙1, q˙2) = 1 2 m1‖q˙1‖2 + 12m2‖q˙1 + q˙2‖ 2 −m1gq1 · k−m2g(q1 + q2) · k. (3.5.10) Note that (3.5.10) has the standard form of kinetic minus potential energy. We identify the velocity vectors q˙1 and q˙2 with vectors perpendicular to q1 and q2, respectively. 68 3. Tangent and Cotangent Bundle Reduction The conjugate momenta are p1 = ∂L ∂q˙1 = m1q˙1 +m2(q˙1 + q˙2) (3.5.11) and p2 = ∂L ∂q˙2 = m2(q˙1 + q˙2) (3.5.12) regarded as vectors in R3 that are paired with vectors orthogonal to q1 and q2 respectively. The Hamiltonian is therefore H(q1,q2,p1,p2) = 1 2m1 ‖p1 − p2‖2 + 12m2 ‖p2‖ 2 +m1gq1 · k +m2g(q1 + q2) · k. (3.5.13) The equations of motion are the Euler–Lagrange equations for L or, equivalently, Hamilton’s equations for H. To write them out explicitly, it is easiest to coordinatize Q. We will describe this process in §5.5. Now let G = S1 act on Q by simultaneous rotation about the z-axis. If Rθ is the rotation by an angle θ, the action is (q1,q2) 7→ (Rθq1, Rθq2). The infinitesimal generator corresponding to the rotation vector ωk is given by ω(k× q1,k× q2) and so the momentum map is 〈J(q1,q2,p1,p2), ωk〉 = ω[p1 · (k× q1) + p2 · (k× q2)] = ωk · [q1 × p1 + q2 × p2] that is, J = k · [q1 × p1 + q2 × p2]. (3.5.14) From (3.5.11) and (3.5.12), J = k · [m1q1 × q˙1 +m2q1 × (q˙1 + q˙2) +m2q2 × (q˙1 + q˙2)] = k · [m1(q1 × q˙1) +m2(q1 + q2)× (q˙1 + q˙2)]. The locked inertia tensor is read off from the metric defining L in (3.5.10): 〈I(q1,q2)ω1k, ω2k〉 = ω1ω2〈〈(k× q1,k× q2), (k× q1,k× q2)〉〉 = ω1ω2{m1‖k× q1‖2 +m2‖k× (q1 + q2)‖2}. Using the identity (k× r) · (k× s) = r · s− (k · r)(k · s), we get I(q1,q2) = m1‖q⊥1 ‖2 +m2‖(q1 + q2)⊥‖2, (3.5.15) 3.5 Examples 69 where ‖q⊥1 ‖2 = ‖q1‖2 − ‖q1 · k‖2 is the square length of the projection of q1 onto the xy-plane. Note that I is the moment of inertia of the system about the k-axis. The mechanical connection is given by (3.3.3): A(q1,q2,v1,v2) = I−1J(m1v1 +m2(v1 + v2),m2(v1 + v2)) = I−1(k · [m1q1 × v1 +m2(q1 + q2)× (v1 + v2)]) = k · [m1q1 × v1 +m2(q1 + q2)× (v1 + v2)] m1‖q⊥1 ‖2 +m2‖(q1 + q2)⊥‖2 . Identifying this linear function of (v1,v2) with a function taking values in R3 × R3 gives A(q1,q2) = 1 m1‖q⊥1 ‖2 +m2‖(q1 + q2)⊥‖2 × [k× ((m1 +m2)q⊥1 +m2q⊥2 ),k×m2(q1 + q2)⊥]. (3.5.16) A short calculation shows that J(α(q1,q2)) = 1, as it should. Also, from (3.5.16), and (3.3.20), Vµ(q1,q2) = m1gq1 · k +m2g(q1 + q2) · k + 1 2 µ2 m1‖q⊥1 ‖2 +m2‖(q1 + q2)⊥‖2 . (3.5.17) The reduced space is T ∗(Q/S1), which is 6-dimensional and it carries a nontrivial magnetic term obtained by taking the differential of (3.5.16). C. The Water (or Ozone) Molecule. We now compute the locked in- ertia tensor, the mechanical connection and the amended potential for this example at symmetric states. Since I(q) : g→ g∗ satisfies 〈I(q)η, ξ〉 = 〈〈ηQ(q), ξQ(q)〉〉 where 〈〈 , 〉〉 is the kinetic energy metric coming from the Lagrangian, I(r, s) is the 3× 3 matrix determined by 〈I(r, s)η, ξ〉 = 〈〈(η × r, η × s), (ξ × r, ξ × s)〉〉. From our expression for the Lagrangian, we get the kinetic energy metric 〈〈(u1,u2), (v1,v2)〉〉 = m2 (u1 · v1) + 2Mm(u2 · v2). Thus, 〈I(r, s)η, ξ〉 = 〈〈(η × r, η × s), (ξ × r, ξ × s)〉〉 = m 2 (η × r) · (ξ × r) + 2Mm(η × s) · (ξ × s) = m 2 ξ · [r× (η × r)] + 2Mmξ · [s× (η × s)] 70 3. Tangent and Cotangent Bundle Reduction and so I(r, s)η = m 2 r× (η × r) + 2Mms× (η × s) = m 2 [η‖r‖2 − r(r · η)] + 2Mm[η‖s‖2 − s(s · η)] = [m 2 ‖r‖2 + 2Mm‖s‖2 ] η − [m 2 r(r · η) + 2Mms(s · η) ] . Therefore, I(r, s) = [m 2 ‖r‖2 + 2Mm‖s‖2 ] Id− [m 2 r⊗ r + 2Mms⊗ s ] , (3.5.18) where Id is the identity. To form the mechanical connection we need to invert I(r, s). To do so, one has to solve[m 2 ‖r‖2 + 2Mm‖s‖2 ] η − [m 2 r(r · η) + 2Mms(s · η) ] = u (3.5.19) for η. An especially easy, but still interesting case is that of symmetric molecules, in which case r and s are orthogonal. Write η in the corresponding orthog- onal basis as: η = ar + bs + c(r× s). (3.5.20) Taking the dot product of (3.5.19) with r, s and r× s respectively, we find a = u · r 2Mm‖r‖2‖s‖2 , b = 2u · s m‖r‖2‖s‖2 , c = u · (r× s) γ‖r‖2‖s‖2 (3.5.21) where γ = γ(r, s) = m 2 ‖r‖2 + 2Mm‖s‖2. With r = ‖r‖ and s = ‖s‖, we arrive at I(r, s)−1 = 1 γ Id + m 4Mms2γ r⊗ r + 4Mm mr2γ s⊗ s = 1 γ [ Id + 1 4Ms2 r⊗ r + 4M r2 s⊗ s ] (3.5.22) We state the general case. Let s⊥ be the projection of s along the line perpendicular to r, so s = s⊥ + s · r r2 r. Then I(r, s)−1u = η, (3.5.23) where η = ar + bs⊥ + cr× s⊥ 3.5 Examples 71 and a = δu · r + βu · s⊥ αδ − β2 , b = βu · r + αu · s⊥ αδ − β2 , c = u · (r× s⊥) γrs⊥ and α = γr2 − 1 2 mr4 − 2Mm(s · r)2, β = 2Mm(s⊥)2s · r, δ = γr2 − 2Mm(s⊥)4. The mechanical connection, defined in general by A : TQ→ g; A(vq) = I(q)−1J(FL(vq)) gives the “overall” angular velocity of the system. For us, let (r, s, r˙, s˙) be a tangent vector and (r, s, pi, σ) the corresponding momenta given by the Legendre transform. Then, A is determined by I · A = r× pi + s× σ or,[m 2 ‖r‖2 + 2Mm‖s‖2 ] A− [m 2 r(r · A) + 2Mms(s · A) ] = r × pi + s× σ. The formula for A is especially simple for symmetric molecules, when r ⊥ s. Then one finds: A = s˙ · n‖s‖ rˆ− r˙ · n ‖r‖ sˆ + 1 γ‖r‖‖s‖ {m 2 ‖r‖2s · r˙− 2Mm‖s‖2r · s˙ } n (3.5.24) where rˆ = r/‖r‖, sˆ = s/‖s‖, and n = (r×s)/‖r‖‖s‖. The formula for a gen- eral r and s is a bit more complicated, requiring the general formula (3.5.23) for I(r, s)−1. Fix a total angular momentum vector µ and let 〈Aµ(q), vq〉 = 〈µ,A(vq)〉, so Aµ(q) ∈ T ∗Q. For example, using A above (for symmetric molecules), we get Aµ = 1 γ [ m 2 (µ× r)− 2Mm r2 (s · µ)(r× s) ] dr + 1 γ [ 2Mm(µ× s) + m 2s2 (r · µ)(r× s) ] ds. The momentum Aµ has the property that J(Aµ) = µ, as may be checked. The amended potential is then Vµ(r, s) = H(Aµ(r, s)). Notice that for the water molecule, the reduced bundle (T ∗Q)µ → T ∗S is an S2 bundle over shape space. The sphere represents the effective “body angular momentum variable”. 72 3. Tangent and Cotangent Bundle Reduction Iwai [1987, 1990] has given nice formulas for the curvature of the mechan- ical connection using a better choice of variables than Jacobi coordinates and Montgomery [1995] has given a related phase formula for the three body problem. 3.6 Lagrangian Reduction and the Routhian So far we have concentrated on the theory of reduction for Hamiltonian systems. There is a similar procedure for Lagrangian systems, although it is not so well known. The Abelian version of this was known to Routh by around 1860 and this is reported in the book of Arnold [1988]. However, that procedure has some difficulties—it is coordinate dependent, and does not apply when the magnetic term is not exact. The procedure developed here (from Marsden and Scheurle [1993a] and Marsden, Ratiu and Scheurle [2000]) avoids these difficulties and extends the method to the nonabelian case. We do this by including conservative gyroscopic forces into the varia- tional principle in the sense of Lagrange and d’Alembert. One uses a Dirac constraint construction to include the cases in which the reduced space is not a tangent bundle (but it is a Dirac constraint set inside one). Some of the ideas of this section are already found in Cendra, Ibort, and Marsden [1987]. The nonabelian case is well illustrated by the rigid body. Given µ ∈ g∗, define the Routhian Rµ : TQ→ R by: Rµ(q, v) = L(q, v)− 〈A(q, v), µ〉 (3.6.1) where A is the mechanical connection. Notice that the Routhian has the form of a Lagrangian with a gyroscopic term; see Bloch, Krishnaprasad, Marsden, and Sanchez de Alvarez [1992] and Wang and Krishnaprasad [1992] for information on the use of gyroscopic systems in control theory. One can also regard Rµ as a partial Legendre transform of L, changing the angular velocity A(q, v) to the angular momentum µ. We will see this explicitly in coordinate calculations below. A basic observation about the Routhian is that solutions of the Euler– Lagrange equations for L can be regarded as solutions of the Euler–Lagrange equations for the Routhian, with the addition of “magnetic forces”. To un- derstand this statement, recall that we define the magnetic two form β to be β = dAµ, (3.6.2) a two form on Q (that drops to Q/Gµ). In coordinates, βij = ∂Aj ∂qi − ∂Ai ∂qj , 3.6 Lagrangian Reduction and the Routhian 73 where we write Aµ = Aidqi and β = ∑ i 0 or −m < α < −1. (4.3.9) The intervals (−∞,m) and (−1, 0) are also possible and correspond to pendulum configurations with the first and second pendulum inverted, re- spectively. Dividing the Equations (4.3.7) to eliminate µ and using a little algebra proves the following result. 4.3.1 Theorem. All of the relative equilibria of the double spherical pen- dulum apart the four equilibria with the two pendula vertical are given by the points on the graph of λ2 = L2 − r2 L2 − α2 (4.3.10) where L(α) = ( 1 + α m )( α 1 + α ) , subject to the restriction 0 ≤ λ2 ≤ r2/α2. From (4.3.7) we can express either µ or ξ in terms of α. In Figures 4.3.1 and 4.3.2 we show the relative equilibria for two sample values of the system parameters. Note that there is a bifurcation of relative equilibria for fixed m and increasing r, and that it occurs within the range of restricted values of α and λ. Also note that there can be two or three relative equilibria for a given set of system parameters. The bifurcation of relative equilibria that happens between Figures 4.3.1 and 4.3.2 does so along the curve in the (r,m) plane given by r = 2m 1 +m as is readily seen. For instance, for m = 2 one gets r = 4/3, in agreement with the figures. The above results are in agreement with those of Baillieul [1987]. 4.3 Examples 93 -3 -2 0.5 0.25 -0.25 -0.5 -1 1 2 0.75 1 1.25 1.5 Figure 4.3.1. The graphs of λ2 verses α for r = 1, m = 2 and of λ2 = r2/α2. -3 -2 0.5 0.25 -0.25 -0.5 -1 1 2 0.75 1 1.25 1.5 Figure 4.3.2. The graph of λ2 verses α for r = 1.35 and m = 2 and of λ2 = r2/α2. C. The water molecule. Because it is a little complicated to work out Vµ (except at symmetric molecules) we will concentrate on the conditions involving Hξ and Vξ for finding the relative equilibria. 94 4. Relative Equilibria Let us work these out in turn. By definition, Hξ = H − 〈J, ξ〉 = 1 4mM ‖σ‖2 + ‖pi‖ 2 m + 1 2M‖J‖ 2 + V − pi · (ξ × r)− σ · (ξ × s). (4.3.11) Thus, the conditions for a critical point associated with δHξ = 0 are: δpi : 2 m pi = ξ × r (4.3.12) δσ : 1 2mM σ = ξ × s (4.3.13) δr : ∂V ∂r = pi × ξ (4.3.14) δs : ∂V ∂s = σ × ξ. (4.3.15) The first two equations express pi and σ in terms of r and s. When they are substituted in the second equations, we get conditions on (r, s) alone: ∂V ∂r = m 2 (ξ × r)× ξ = m 2 [ ξ(ξ · r)− r‖ξ‖2] (4.3.16) and ∂V ∂s = 2mM(ξ × s)× ξ = 2mM [ξ(ξ · s)− s‖ξ‖2] . (4.3.17) These are the conditions for a relative equilibria we sought. Any collection (ξ, r, s) satisfying (4.3.16) and (4.3.17) gives a relative equilibrium. Now consider Vξ for the water molecule: Vξ(r, s) = V (r, s)− 12 [ ‖ξ‖2 (m 2 ‖r‖2 + 2Mm‖s‖2 )] − 1 2 [m 2 (ξ · r)2 + 2Mm(ξ · s)2 ] . (4.3.18) The conditions for a critical point are: δr : ∂V ∂r − ‖ξ‖2mr 2 − m 2 (ξ · r)ξ = 0 (4.3.19) δs : ∂V ∂s − ‖ξ‖2 · 2Mms− 2Mm(ξ · s)ξ = 0, (4.3.20) which coincide with (4.3.16) and (4.3.17). The extra condition from Propo- sition 4.2.1, namely pe = FL(ξQ(qe)), gives (4.3.12) and (4.3.13). 4.4 The Rigid Body 95 4.4 The Rigid Body Since examples like the water molecule have both rigid and internal vari- ables, it is important to understand our constructions for the rigid body itself. The rotation group SO(3) consists of all orthogonal linear transforma- tions of Euclidean three space to itself, which have determinant one. Its Lie algebra, denoted so(3), consists of all 3 × 3 skew matrices, which we identify with R3 by the isomorphism ˆ: R3 → so(3) defined by Ω 7→ Ωˆ =  0 −Ω3 Ω2Ω3 0 −Ω1 −Ω2 Ω1 0  , (4.4.1) where Ω = (Ω1,Ω2,Ω3). One checks that for any vectors r, and Θ, Ωˆr = Ω× r, and ΩˆΘˆ− ΘˆΩˆ = (Ω×Θ)ˆ. (4.4.2) Equations (4.4.1) and (4.4.2) identify the Lie algebra so(3) with R3 and the Lie algebra bracket with the cross product of vectors. If Λ ∈ SO(3) and Θˆ ∈ so(3), the adjoint action is given by AdΛΘˆ = [ΛΘ]ˆ . (4.4.3) The fact that the adjoint action is a Lie algebra homomorphism, corre- sponds to the identity Λ(r× s) = Λr× Λs, (4.4.4) for all r, s ∈ R3. Given Λ ∈ SO(3), let vˆΛ denote an element of the tangent space to SO(3) at Λ. Since SO(3) is a submanifold of GL(3), the general linear group, we can identify vˆΛ with a 3× 3 matrix, which we denote with the same letter. Linearizing the defining (submersive) condition ΛΛT = 1 gives ΛvˆTΛ + vˆΛΛ T = 0, (4.4.5) which defines TΛ SO(3). We can identify TΛ SO(3) with so(3) by two iso- morphisms: First, given Θˆ ∈ so(3) and Λ ∈ SO(3), define (Λ, Θˆ) 7→ ΘˆΛ ∈ TΛ SO(3) by letting ΘˆΛ be the left invariant extension of Θˆ: ΘˆΛ := TeLΛ · Θˆ ∼= (Λ,ΛΘˆ). (4.4.6) Second, given θˆ ∈ so(3) and Λ ∈ SO(3), define (Λ, θˆ) 7→ θˆΛ ∈ TΛ SO(3) through right translations by setting θˆΛ := TeRΛ · θˆ ∼= (Λ, θˆΛ). (4.4.7) The notation is partially dictated by continuum mechanics considerations; upper case letters are used for the body (or convective) variables and lower 96 4. Relative Equilibria case for the spatial (or Eulerian) variables. Often, the base point is omit- ted and with an abuse of notation we write ΛΘˆ and θˆΛ for ΘˆΛ and θˆΛ, respectively. The dual space to so(3) is identified with R3 using the standard dot product : Π · Θ = 12 tr[ΠˆT Θˆ]. This extends to the left-invariant pairing on TΛ SO(3) given by 〈ΠˆΛ, ΘˆΛ〉 = 12tr[Πˆ T ΛΘˆΛ] = 1 2 tr[ΠˆT Θˆ] = Π ·Θ. (4.4.8) Write elements of so(3)∗ as Πˆ, where Π ∈ R3, (or pˆi with pi ∈ R3) and elements of T ∗Λ SO(3) as ΠˆΛ = (Λ,ΛΠˆ), (4.4.9) for the body representation, and for the spatial representation pˆiΛ = (Λ, pˆiΛ). (4.4.10) Again, explicit indication of the base point will often be omitted and we shall simply write ΛΠˆ and pˆiΛ for ΠˆΛ and pˆiΛ, respectively. If (4.4.9) and (4.4.10) represent the same covector, then pˆi = ΛΠˆΛT , (4.4.11) which coincides with the co-adjoint action. Equivalently, using the isomor- phism (4.4.2) we have pi = ΛΠ. (4.4.12) The mechanical set-up for rigid body dynamics is as follows: the config- uration manifold Q and the phase space P are Q = SO(3) and P = T ∗ SO(3) (4.4.13) with the canonical symplectic structure. The Lagrangian for the free rigid body is, as in Chapter 1, its kinetic energy: L(Λ, Λ˙) = 1 2 ∫ B ρref(X)‖Λ˙X‖2 d3X (4.4.14) where ρref is a mass density on the reference configuration B. The La- grangian is evidently left invariant. The corresponding metric is the left invariant metric given at the identity by 〈〈ξˆ, ηˆ〉〉 = ∫ B ρref(X) ξˆ(X) · ηˆ(X) d3X or on R3 by 〈〈a, b〉〉 = ∫ B ρref(X)(a×X) · (b×X) d3X = a · Ib (4.4.15) 4.4 The Rigid Body 97 where I = ∫ B ρref(X)[‖X‖2Id−X ⊗X] d3X (4.4.16) is the inertia tensor . The corresponding Hamiltonian is H = 1 2 Π · I−1Π = 1 2 pi · I−1pi (4.4.17) where I = ΛIΛ−1 is the time dependent inertia tensor . The expression H = 12Π ·I−1Π reflects the left invariance of H under the action of SO(3). Thus left reduction by SO(3) to body coordinates induces a function on the quotient space T ∗ SO(3)/ SO(3) ∼= so(3)∗. The symplectic leaves are spheres, ‖Π‖ = constant. The induced function on these spheres is given by (4.4.17) regarded as a function of Π. The dynamics on this sphere is obtained by intersection of the sphere ‖Π‖2 = constant and the ellipsoid H = constant that we discussed in Chapter 1. Consistent with the preceding discussion, let G = SO(3) act from the left on Q = SO(3); that is, Q · Λ = LQΛ = QΛ, (4.4.18) for all Λ ∈ SO(3) and Q ∈ G. Hence the action of G = SO(3) on P = T ∗ SO(3) is by cotangent lift of left translations. Since the infinitesimal generator associated with ξˆ ∈ so(3) is obtained as ξˆSO(3)(Λ) = d dt exp [ tξˆ ] Λ ∣∣∣∣ t=0 = ξˆΛ, (4.4.19) the corresponding momentum map is〈 J(pˆiλ), ξˆ 〉 = 1 2 tr [ pˆiTΛξso(3)Λ ] = 1 2 tr [ ΛT pˆiT ξˆΛ ] = 1 2 tr [ pˆiT ξˆ ] = pi · ξ, (4.4.20) that is, J(pˆiΛ) = pˆi, or J(ξˆ) = pi · ξ. (4.4.21) To locate the relative equilibria, consider Hξ(pˆiΛ) = H − [(J(ξ)− pie) · ξ] = 12pi · I −1pi − ξ · (pi − pie). (4.4.22) To find its critical points, we note that although pˆiΛ ∈ T ∗ SO(3) are the basic variables, it is more convenient to regard Hξ as a function of (Λ, pi) ∈ SO(3)× R3∗ through the isomorphism (4.4.10). Thus, let pˆiΛe ∼= (Λe, pˆieΛe) ∈ T ∗ SO(3) be a relative equilibrium point. For any δθ ∈ R3 we construct the curve 7→ Λ = exp [ δ̂θ ] Λe ∈ SO(3), (4.4.23) 98 4. Relative Equilibria which starts at Λe and d d Λ ∣∣∣∣ =0 = δ̂θΛ. (4.4.24) Let δpi ∈ (R3)∗ and consider the curve in (R3)∗ defined as 7→ pi = pie + δpi ∈ (R3)∗, (4.4.25) which starts at pie. These constructions induce a curve 7→ pˆiΛe ∈ T ∗ SO(3) via the isomorphism (4.4.10); that is, pˆiΛe := (Λ, pˆiΛ). With this notation at hand we compute the first variation using the chain rule. Let δHξ |e:= d d Hξ, ∣∣∣∣ =0 = 0, (4.4.26) where Hξ, := 1 2 pi · I−1 pi − ξ · pi and I−1 := ΛI−1ΛT . In addition, at equilibrium, J(ze) = µ reads pi = pie. (4.4.27) To compute δHξ, observe that 1 2 pie · d d I−1 pie ∣∣∣∣ =0 = 1 2 pie · [ δθˆI−1e − I−1e δθˆ ] pie = 1 2 [ pie · δθ × I−1e pie − I−1e pie · δθ × pie ] = δθ · (I−1e pie × pie) , (4.4.28) where we have made use of elementary vector product identities. Thus, δHξ |e= δpi · [ I−1e pie − ξ ] + δθ · [I−1e pie × pie] = 0. (4.4.29) From this relation we obtain the two equilibrium conditions: I−1e pie × pie = 0 and I−1e pie = ξ. (4.4.30) Equivalently, ξ × pie = 0 and I−1e ξ = λξ, (4.4.31) where λ > 0 by positive definiteness of Ie = ΛeIΛTe . These conditions state that pie is aligned with a principal axis, and that the rotation is about this axis. Note that pie = Ieξ, so that ξ does indeed correspond to the angular velocity. Equivalently, we can locate the relative equilibria using Proposition 4.2.1. The locked inertia tensor (as the notation suggests) is just I and here 4.4 The Rigid Body 99 pe = FL(ξQ(qe)) reads pie = Ieξ, while qe = Λe is required to be a critical point of Vξ or Vµ, where µ = pie = Iξ. Thus Vξ(Λ) = 1 2 ξ · (Iξ) and Vµ(Λ) = −12µ · (I −1µ). (4.4.32) The calculation giving (4.4.28) shows that Λe is a critical point of Vξ, or of Vµ iff ξ × pie = 0. These calculations show that rigid body relative equilibria correspond to steady rotational motions about their principal axes. The “global” per- spective taken above is perhaps a bit long winded, but it is a point of view that is useful in the long run. 100 4. Relative Equilibria Page 101 5 The Energy–Momentum Method This chapter develops the energy–momentum method of Simo, Posbergh, and Marsden [1990, 1991] and Simo, Lewis, and Marsden [1991]. This is a technique for determining the stability of relative equilibria and for putting the equations of motion linearized at a relative equilibrium into normal form. This normal form is based on a special decomposition of vari- ations into rigid and internal components that gives a block structure to the Hamiltonian and symplectic structure. There has been considerable de- velopment of stability and bifurcation techniques over the last decade, and some properties like block diagonalization have been seen in a variety of problems; for example, this appears to be what is happening in Morrison and Pfirsch [1990]. 5.1 The General Technique In these lectures we will confine ourselves to the regular case ; that is, we assume ze is a relative equilibrium that is also a regular point (i.e., gze = {0}, or ze has a discrete isotropy group) and µ = J(ze) is a generic point in g∗ (i.e., its orbit is of maximal dimension). We are seeking condi- tions for Gµ-orbital stability of ze; that is, conditions under which for any neighborhood U of the orbit Gµ · ze, there is a neighborhood V with the property that the trajectory of an initial condition starting in V remains in U for all time. To do so, find a subspace S ⊂ TzeP 102 5. The Energy–Momentum Method satisfying two conditions (see Figure 5.1.1): (i) S ⊂ ker DJ(ze) and (ii) S is transverse to the Gµ-orbit within ker DJ(ze). The energy–momentum method is as follows: (a) find ξ ∈ g such that δHξ(ze) = 0, (b) test δ2Hξ(ze) for definiteness on S. S Figure 5.1.1. The energy–momentum method tests the second variation of the augmented Hamiltonian for definiteness on the space S. 5.1.1 Theorem (The Energy–Momentum Theorem). If δ2Hξ(ze) is def- inite, then ze is Gµ-orbitally stable in J−1(µ) and G-orbitally stable in P . Proof. First, one has Gµ-orbital stability within J−1(µ) because the re- duced dynamics induces a well defined dynamics on the orbit space Pµ; this dynamics induces a dynamical system on S for which Hξ is an invariant function. Since it has a non-degenerate extremum at ze, invariance of its level sets gives the required stability. Second, one gets Gν orbital stability within J−1(ν) for ν close to µ since the form of Figure 5.1.1 changes in a regular way for nearby level sets, as µ is both a regular value and a generic point in g∗. For results for non-generic µ or non-regular µ, see Patrick [1990] and Lewis [1992]. 5.1 The General Technique 103 A well known example that may be treated as an instance of the energy– momentum method, where the necessity of considering orbital stability is especially clear, is the stability of solitons in the KdV equation. (A result of Benjamin [1972] and Bona [1975].) Here one gets stability of solitons modulo translations; one cannot expect literal stability because a slight change in the amplitude cases a translational drift, but the soliton shape remains dynamically stable. This example is actually infinite dimensional, and here one must employ additional hypotheses for Theorem 5.1.1 to be valid. Two possible infinite dimensional versions are as follows: one uses convexity hypotheses going back to Arnold [1969] (see Holm, Kupershmidt, and Levermore [1985] for a concise summary) or, what is appropriate for the KdV equation, em- ploy Sobolev spaces on which the calculus argument of Theorem 5.1.1 is correct—one needs the energy norm defined by δ2Hξ(ze) to be equivalent to a Sobolev norm in which one has global existence theorems and on which Hξ is a smooth function. Sometimes, such as in three dimensional elasticity, this can be a serious difficulty (see, for instance, Ball and Marsden [1984]). In other situations, a special analysis is needed, as in Wan and Pulvirente [1984] and Batt and Rein [1993]. The energy–momentum method can be compared to Arnold’s energy– Casimir method that takes place on the Poisson manifold P/G rather than on P itself. Consider the diagram in Figure 5.1.2. P/G g∗ P R C Φ pi J @ @ @ @R @ @ @ @R Figure 5.1.2. Comparing the energy–Casimir and energy–momentum methods. The energy–Casimir method searches for a Casimir function C on P/G such that H + C has a critical point at ue = [ze] for the reduced system and then requires definiteness of δ2(H+C)(ue). If one has a relative equilibrium ze (with its associated µ and ξ), and one can find an invariant 104 5. The Energy–Momentum Method function Φ on g∗ such that δΦ δµ (µ) = −ξ, then one can define a Casimir C by C ◦ pi = Φ ◦ J and H + C will have a critical point at ue. Remark. This diagram gives another way of viewing symplectic reduc- tion, namely, as the level sets of Casimir functions in P/G. In general, the symplectic reduced spaces may be viewed as the symplectic leaves in P/G. These may or may not be realizable as level sets of Casimir functions. The energy–momentum method is more powerful in that it does not depend on the existence of invariant functions, which is a serious difficulty in some examples, such as geometrically exact elastic rods, certain plasma problems, and 3-dimensional ideal flow. On the other hand, the energy– Casimir method is able to treat some singular cases rather easily since P/G can still be a smooth manifold in the singular case. However, there is still the problem of interpretation of the results in P ; see Patrick [1990]. For simple mechanical systems, one way to choose S is as follows. Let V = { δq ∈ TqeQ | 〈〈δq, χQ(qe)〉〉 = 0 for all χ ∈ gµ }, that is, the metric orthogonal of the tangent space to the Gµ-orbit in Q. Let S = { δz ∈ ker dJ(ze) | TpiQ · δz ∈ V } where piQ : T ∗Q = P → Q is the projection. The following gauge invariance condition will be helpful. 5.1.2 Proposition. Let ze ∈ P be a relative equilibrium, and let G · ze = { g · ze | g ∈ G } be the orbit through ze with tangent space Tze(G · ze) = { ηP (ze) | η ∈ g }. (5.1.1) Then, for any δz ∈ Tze [J−1(µ)], we have δ2Hξ(ze) · (ηP (ze), δz) = 0 for all η ∈ g. (5.1.2) Proof. Since H : P → R is G-invariant, Ad∗-equivariance of the momen- tum map gives Hξ(g · z) = H(g · z)− 〈J(g · z), ξ〉+ 〈µ, ξ〉 = H(z)− 〈Ad∗g−1(J(z)), ξ〉+ 〈µ, ξ〉 = H(z)− 〈J(z),Adg−1(ξ)〉+ 〈µ, ξ〉, (5.1.3) 5.2 Example: The Rigid Body 105 for any g ∈ G and z ∈ P . Choosing g = exp(tη) with η ∈ g, and differenti- ating (5.1.3) with respect to t gives dHξ(z) · ηP (z) = − 〈 J(z), d dt ∣∣∣∣ t=0 Adexp(−tη)(ξ) 〉 = 〈J(z), [η, ξ]〉. (5.1.4) Taking the variation of (5.1.4) with respect to z, evaluating at ze and using the fact that dHξ(ze) = 0, one gets the expression δ2Hξ(ze)(ηP (ze), δz) = 〈TzeJ · δz, [η, ξ]〉, (5.1.5) which vanishes if TzeJ(ze) · δz = 0, that is, if δz ∈ ker [TzeJ(ze)] = TzeJ −1(µe). In particular, from the above result and Proposition 4.1.3, we have 5.1.3 Proposition. δ2Hξ(ze) vanishes identically on ker [TzeJ(ze)] along the directions tangent to the orbit Gµ · ze; that is δ2Hξ(ze) · (ηP (ze), ζP (ze)) = 0 for any η, ζ ∈ gµ. (5.1.6) Proof. A general fact about reduction is that Tze(Gµ · ze) = Tze(G · ze)∩ ker[TzeJ(ze)]. Since Tze(Gµ ·ze) ⊂ Tze(G ·ze) the result follows from (5.1.2) by taking δz = ξP (ze) with ξ ∈ gµ. These propositions confirm the general geometric picture set out in Fig- ure 5.1.1. They show explicitly that the orbit directions are neutral direc- tions of δ2Hξ(ze), so one has no chance of proving definiteness except on a transverse to the Gµ-orbit. Remark. Failure to understand these simple distinctions between sta- bility and orbital stability can sometimes lead to misguided assertions like “Arnold’s method can only work at symmetric states”. (See Andrews [1984], Chern and Marsden [1990] and Carnevale and Shepherd [1990].) 5.2 Example: The Rigid Body The classical result about free rigid body stability mentioned in Chapter 1 is that uniform rotation about the longest and shortest principal axes are stable motions, while motion about the intermediate axis is unstable. A simple way to see this is by using the energy–Casimir method. We begin with the equations of motion in body representation: Π˙ = dΠ dt = Π× Ω (5.2.1) 106 5. The Energy–Momentum Method where Π,Ω ∈ R3, Ω is the angular velocity, and Π is the angular momentum. Assuming principal axis coordinates, Πj = IjΩj for j = 1, 2, 3, where I = (I1, I2, I3) is the diagonalized moment of inertia tensor, I1, I2, I3 > 0. As we saw in Chapter 2, this system is Hamiltonian in the Lie–Poisson structure with the kinetic energy Hamiltonian H(Π) = 1 2 Π · Ω = 1 2 3∑ i=0 Π2i Ii , (5.2.2) and that for a smooth function Φ : R→ R, the function CΦ(Π) = Φ ( 1 2 ‖Π‖2 ) (5.2.3) is a Casimir function. We first choose Φ such that HCΦ := H + CΦ has a critical point at a given equilibrium point of (5.2.1). Such points occur when Π is parallel to Ω. We can assume that the equilibrium solution is Πe = (1, 0, 0). The derivative of HCΦ(Π) = 1 2 3∑ i=0 Π2i Ii + Φ ( 1 2 ‖Π‖2 ) is dHCΦ(Π) · δΠ = ( Ω + ΠΦ′ ( 1 2 ‖Π‖2 )) · δΠ. (5.2.4) This equals zero at Π = (1, 0, 0), provided that Φ′ ( 1 2 ) = − 1 I1 . (5.2.5) The second derivative at the equilibrium Πe = (1, 0, 0) is d2HCΦ(Πe) · (δΠ, δΠ) = δΩ · δΠ + Φ′ ( 1 2 ‖Πe‖2 ) ‖δΠ‖2 + (Πe · δΠ)2Φ′′ ( 1 2 ‖Πe‖2 ) = 3∑ i=0 (δΠi)2 Ii − ‖δΠ‖ 2 I1 + Φ′′ ( 1 2 ) (δΠ1)2 = ( 1 I2 − 1 I1 ) (δΠ2)2 + ( 1 I3 − 1 I1 ) (δΠ3)2 + Φ′′ ( 1 2 ) (δΠ1)2. (5.2.6) This quadratic form is positive definite if and only if Φ′′ ( 1 2 ) > 0 (5.2.7) 5.2 Example: The Rigid Body 107 and I1 > I2, I1 > I3. (5.2.8) Consequently, Φ(x) = − ( 1 I1 ) x+ ( x− 1 2 )2 satisfies (5.2.5) and makes the second derivative of HCΦ at (1, 0, 0) posi- tive definite, so stationary rotation around the longest axis is stable. The quadratic form is negative definite provided Φ′′ ( 1 2 ) < 0 (5.2.9) and I1 < I2, I1 < I3. (5.2.10) A specific function Φ satisfying the requirements (5.2.5) and (5.2.9) is Φ(x) = − ( 1 I1 ) x− ( x− 1 2 )2 . This proves that the rigid body in steady rotation around the short axis is (Liapunov) stable. Finally, the quadratic form (5.2.6) is indefinite if I1 > I2, I3 > I1. (5.2.11) or the other way around. One needs an additional argument to show that rotation around the middle axis is unstable. Perhaps the simplest way is as follows: Linearizing equation (5.2.1) at Πe = (1, 0, 0) yields the linear constant coefficient system d dt δΠ = δΠ× Ωe + Πe × δΩ = ( 0, I3 − I1 I3I1 δΠ3, I1 − I2 I1I2 δΠ2 ) =  0 0 0 0 0 I3 − I1 I3I1 0 I1 − I2 I1I2 0  δΠ. (5.2.12) On the tangent space at Πe to the sphere of radius ‖Πe‖ = 1, the linear operator given by this linearized vector field has matrix given by the lower right 2× 2 block, whose eigenvalues are ± 1 I1 √ I2I3 √ (I1 − I2)(I3 − I1). Both eigenvalues are real by (5.2.11) and one is strictly positive. Thus Πe is spectrally unstable and thus is (nonlinearly) unstable. Thus, in the motion 108 5. The Energy–Momentum Method of a free rigid body, rotation around the long and short axes is (Liapunov) stable and around the middle axis is unstable. The energy–Casimir method deals with stability for the dynamics on Π- space. However, one also wants (corresponding to what one “sees”) to show orbital stability for the motion in T ∗ SO(3). This follows from stability in the variable Π and conservation of spatial angular momentum. The energy– momentum method gives this directly. While it appears to be more complicated for the rigid body, the power of the energy–momentum method is revealed when one does more complex examples, as in Simo, Posbergh, and Marsden [1990] and Lewis and Simo [1990]. In §4.4 we set up the rigid body as a mechanical system on T ∗ SO(3) and we located the relative equilibria. By differentiating as in (4.4.29) we find the following formula: δ2Hξ|e((δpi, δθ), (δpi, δθ)) = [δpiT δθT ]  I−1e (I−1e − λ1)pˆie −pˆie(I−1e − λ1) −pˆie(I−1e − λ1)pˆie  δpi δθ  . (5.2.13) Note that this matrix is 6×6. Corresponding to the two conditions on S, we restrict the admissible variations (δ, pi, δθ) ∈ R3∗ × R3. Here, J(pˆi∧) = pˆiΛ; hence µ = pˆie and Tze(Gµ · ze) = infinitesimal rotations about the axis pie; that is, multiples of pie, or equivalently ξ. Variations that are orthogonal to this space and also lie in the space δpi = 0 (which is the condition δJ = δpi = 0) are of the form δθ with δθ ⊥ pie. Thus, we choose S = { (δpi, δθ) | δpi = 0, δθ ⊥ pie }. (5.2.14) Note that δθ is a variation that rotates pie on the sphere Opie := {pi ∈ R3 | ‖pi‖2 = ‖pie‖2 }, (5.2.15) which is the co-adjoint orbit through pie. The second variation (5.2.13) restricted to the subspace S is given by δ2Hξ,µ|e = δθ · ( pˆiTe (I−1e − λ1)pˆie ) δθ = (pie × δθ) · (I−1e − λ1)(pie × δθ). (5.2.16) If λ is the largest or smallest eigenvalue of I, (5.2.16) will be definite; note that the null space of I−1 − λ1 in (5.2.16) consists of vectors parallel to pie, which have been excluded. Also note that in this example, S is a 2- dimensional space and (5.2.16) in fact represents a 2 × 2 matrix. As we shall see below, this 2 × 2 block can also be viewed as δ2Vµ(qe) on the space of rigid variations. 5.3 Block Diagonalization 109 5.3 Block Diagonalization If the energy–momentum method is applied to mechanical systems with Hamiltonian H of the form kinetic energy (K) plus potential (V ), it is pos- sible to choose variables in a way that makes the determination of stability conditions sharper and more computable. In this set of variables (with the conservation of momentum constraint and a gauge symmetry constraint imposed on S), the second variation of δ2Hξ block diagonalizes; schemati- cally, δ2Hξ =  [ 2× 2 rigid body block ] 0 0 [ Internal vibration block ]  . Furthermore, the internal vibrational block takes the form[ Internal vibration block ] = [ δ2Vµ 0 0 δ2Kµ ] where Vµ is the amended potential defined earlier, and Kµ is a momentum shifted kinetic energy, so formal stability is equivalent to δ2Vµ > 0 and that the overall structure is stable when viewed as a rigid structure, which, as far as stability is concerned, separates the overall rigid body motions from the internal motions of the system under consideration. The dynamics of the internal vibrations (such as the elastic wave speeds) depend on the rotational angular momentum. That is, the internal vibra- tional block is µ-dependent, but in a way we shall explicitly calculate. On the other hand, these two types of motions do not dynamically decouple, since the symplectic form does not block diagonalize. However, the sym- plectic form takes on a particularly simple normal form as we shall see. This allows one to put the linearized equations of motion into normal form as well. To define the rigid-internal splitting, we begin with a splitting in con- figuration space. Consider (at a relative equilibrium) the space V defined above as the metric orthogonal to gµ(q) in TqQ. Here we drop the subscript e for notational convenience. Then we split V = VRIG ⊕ VINT (5.3.1) as follows. Define VRIG = { ηQ(q) ∈ TqQ | η ∈ g⊥µ } (5.3.2) where g⊥µ is the orthogonal complement to gµ in g with respect to the locked inertia metric. (This choice of orthogonal complement depends on q, but we do not include this in the notation.) From (3.3.1) it is clear that 110 5. The Energy–Momentum Method VRIG ⊂ V and that VRIG has the dimension of the coadjoint orbit through µ. Next, define VINT = { δq ∈ V | 〈η, [dI(q) · δq] · ξ〉 = 0 for all η ∈ g⊥µ } (5.3.3) where ξ = I(q)−1µ. An equivalent definition is VINT = { δq ∈ V | [dI(q)−1 · δq] · µ ∈ gµ }. The definition of VINT has an interesting mechanical interpretation in terms of the objectivity of the centrifugal force in case G = SO(3); see Simo, Lewis, and Marsden [1991]. Define the Arnold form Aµ : g⊥µ × g⊥µ → R by Aµ(η, ζ) = 〈ad∗ηµ, χ(q,µ)(ζ)〉 = 〈µ, adηχ(q,µ)(ζ)〉, (5.3.4) where χ(q,µ) : g⊥µ → g is defined by χ(q,µ)(ζ) = I(q)−1ad∗ζµ+ adζI(q)−1µ. The Arnold form appears in Arnold [1966] stability analysis of relative equilibria in the special case Q = G. At a relative equilibrium, the form Aµ is symmetric, as is verified either directly or by recognizing it as the second variation of Vµ on VRIG × VRIG (see (5.3.15) below for this calculation). At a relative equilibrium, the form Aµ is degenerate as a symmetric bilinear form on g⊥µ when there is a non-zero ζ ∈ g⊥µ such that I(q)−1ad∗ζµ+ adζI(q)−1µ ∈ gµ; in other words, when I(q)−1 : g∗ → g has a nontrivial symmetry relative to the coadjoint-adjoint action of g (restricted to g⊥µ ) on the space of linear maps from g∗ to g. (When one is not at a relative equilibrium, we say the Arnold form is non-degenerate when Aµ(η, ζ) = 0 for all η ∈ g⊥µ implies ζ = 0.) This means, for G = SO(3) that Aµ is non-degenerate if µ is not in a multidimensional eigenspace of I−1. Thus, if the locked body is not symmetric (i.e., a Lagrange top), then the Arnold form is non-degenerate. 5.3.1 Proposition. If the Arnold form is non-degenerate, then V = VRIG ⊕ VINT. (5.3.5) Indeed, non-degeneracy of the Arnold form implies VRIG ∩ VINT = {0} and, at least in the finite dimensional case, a dimension count gives (5.3.5). In the infinite dimensional case, the relevant ellipticity conditions are needed. The split (5.3.5) can now be used to induce a split of the phase space S = SRIG ⊕ SINT. (5.3.6) 5.3 Block Diagonalization 111 Using a more mechanical viewpoint, Simo, Lewis, and Marsden [1991] show how SRIG can be defined by extending VRIG from positions to momenta using superposed rigid motions. For our purposes, the important character- ization of SRIG is via the mechanical connection: SRIG = Tqαµ · VRIG, (5.3.7) so SRIG is isomorphic to VRIG. Since αµ maps Q to J−1(µ) and VRIG ⊂ V, we get SRIG ⊂ S. Define SINT = { δz ∈ S | δq ∈ VINT }; (5.3.8) then (5.3.6) holds if the Arnold form is non-degenerate. Next, we write SINT =WINT ⊕W∗INT, (5.3.9) where WINT and W∗INT are defined as follows: WINT = Tqαµ · VINT and W∗INT = { vert(γ) | γ ∈ [g · q]0 } (5.3.10) where g ·q = { ζQ(q) | ζ ∈ g }, [g ·q]0 ⊂ T ∗qQ is its annihilator, and vert(γ) ∈ Tz(T ∗Q) is the vertical lift of γ ∈ T ∗qQ; in coordinates, vert(qi, γj) = (qi, pj , 0, γj). The vertical lift is given intrinsically by taking the tangent to the curve σ(s) = z + sγ at s = 0. 5.3.2 Theorem (Block Diagonalization Theorem). In the splittings in- troduced above at a relative equilibrium, δ2Hξ(ze) and the symplectic form Ωze have the following form: δ2Hξ(ze) =  [ Arnold form ] 0 0 0 δ2Vµ 0 0 0 δ2Kµ  and Ωze =  [ coadjoint orbit symplectic form ] [ internal rigid coupling ] 0 − [ internal rigid coupling ] S 1 0 −1 0  where the columns represent elements of SRIG, WINT, and W∗INT, respec- tively. Remarks. 112 5. The Energy–Momentum Method 1. For the viewpoint of these splittings being those of a connection, see Lewis, Marsden, Ratiu, and Simo [1990]. 2. Arms, Fischer, and Marsden [1975] and Marsden [1981] describe an important phase space splitting of TzP at a point z ∈ J−1(0) into three pieces, namely TzP = Tz(G · z)⊕ S ⊕ [TzJ−1(0)]⊥, where S is, as above, a complement to the gauge piece Tz(G·z) within TzJ−1(0) and can be taken to be the same as S defined in §5.1, and [TzJ−1(0)]⊥ is a complement to TzJ−1(0) within TzP . This splitting has profound implications for general relativistic fields, where it is called the Moncrief splitting . See, for example, Arms, Marsden, and Moncrief [1982] and references therein. Thus, we get, all together, a six-way splitting of TzP . It would be of interest to explore the geometry of this situation further and arrive at normal forms for the linearized dynamical equations valid on all of TzP . 3. It is also of interest to link the normal forms here with those in singularity theory. In particular, can one use the forms here as first terms in higher order normal forms? The terms appearing in the formula for Ωze will be explained in §5.4 below. We now give some of the steps in the proof. 5.3.3 Lemma. δ2Hξ(ze) · (∆z, δz) = 0 for ∆z ∈ SRIG and δz ∈ SINT. Proof. In Chapter 3 we saw that for z ∈ J−1(µ), Hξ(z) = Kµ(z) + Vµ(q) (5.3.11) where each of the functions Kµ and Vµ has a critical point at the rela- tive equilibrium. It suffices to show that each term separately satisfies the lemma. The second variation of Kµ is δ2Kµ(ze) · (∆z, δz) = 〈〈∆p− Tαµ(qe) ·∆q, δp− Tαµ(qe) · δq〉〉. (5.3.12) By (5.3.7), ∆p = Tαµ(qe) · ∆q, so this expression vanishes. The second variation of Vµ is δ2Vµ(qe) · (∆q, δq) = δ(δV (q) ·∆q + 12µ[dI −1(q) ·∆q]µ) · δq. (5.3.13) By G-invariance, δV (q)·∆q is zero for any q ∈ Q, so the first term vanishes. As for the second, let ∆q = ηQ(q) where η ∈ g⊥µ . As was mentioned in Chapter 3, I is equivariant: 〈I(gq)Adgχ,Adgζ〉 = 〈I(q)χ, ζ〉. 5.3 Block Diagonalization 113 Differentiating this with respect to g at g = e in the direction η gives (dI(q) ·∆q)χ+ I(q)adηχ+ ad∗η[I(q)χ] = 0 and so dI−1(q) ·∆q ν = −I(q)−1dI(q) ·∆qI(q)−1ν = adηI(q)−1ν + I(q)−1ad∗ην. (5.3.14) Differentiating with respect to q in the direction δq and letting ν = µ gives d2I−1(q) · (∆q, δq)] · µ = adη[(DI(q)−1 · δq)µ] + (DI(q)−1 · δq) · ad∗ηµ. Pairing with µ and using symmetry of I gives 〈µ,D2I−1(q) · (∆q, δq) · µ〉 = 2〈µ, adη[(DI(q)−1 · δq)µ]〉. (5.3.15) If δq is a rigid variation, we can substitute (5.3.14) in (5.3.15) to express everything using I itself. This is how one gets the Arnold form in (5.3.4). If, on the other hand, δq ∈ VINT, then ζ := (DI(q)−1δq)µ ∈ gµ, so (5.3.15) gives δ2Vµ(qe) · (∆q, δq) = 〈µ, adηζ〉 = −〈ad∗ζµ, η〉 which vanishes since ζ ∈ gµ. These calculations and similar ones given below establish the block diag- onal structure of δ2Hξ(ze). We shall see in Chapter 8 that discrete symme- tries can produce interesting subblocking within δ2Vµ on VINT. Lewis [1992] has shown how to perform these same constructions for general Lagrangian systems. This is important because not all systems are of the form kinetic plus potential energy. For example, gyroscopic control systems are of this sort. We shall see one example in Chapter 7 and refer to Wang and Krish- naprasad [1992] for others, some of the general theory of these systems and some control theoretic applications. The hallmark of gyroscopic systems is in fact the presence of magnetic terms and we shall discuss this next in the block diagonalization context. As far as stability is concerned, we have the following consequence of block diagonalization. 5.3.4 Theorem (Reduced Energy–Momentum Method). Let ze = (qe, pe) be a cotangent relative equilibrium and assume that the internal variables are not trivial; that is, VINT 6= {0}. If δ2Hξ(ze) is definite, then it must be positive definite. Necessary and sufficient conditions for δ2Hξ(ze) to be positive definite are: 1. The Arnold form is positive definite on VRIG and 2. δ2Vµ(qe) is positive definite on VINT. 114 5. The Energy–Momentum Method This follows from the fact that δ2Kµ is positive definite and δ2Hξ has the above block diagonal structure. In examples, it is this form of the energy–momentum method that is often the easiest to use. Finally in this section we should note the relation between δ2Vξ(qe) and δ2Vµ(qe). A straightforward calculation shows that δ2Vµ(qe) · (δq, δq) = δ2Vξ(qe) · (δq, δq) + 〈I(qe)−1dI(qe) · δqξ,dI(qe) · δqξ〉 (5.3.16) and the correction term is positive. Thus, if δ2Vξ(qe) is positive definite, then so is δ2Vµ(qe), but not necessarily conversely. Thus, δ2Vµ(qe) gives sharp conditions for stability (in the sense of predicting definiteness of δ2Hξ(ze)), while δ2Vξ gives only sufficient conditions. Using the notation ζ = [DI−1(qe) · δq]µ ∈ gµ (see (5.3.3)), observe that the “correcting term” in (5.3.16) is given by 〈I(qe)ζ, ζ〉 = 〈〈ζQ(qe), ζQ(qe)〉〉. Formula (5.3.16) is often the easiest to compute with since, as we saw with the water molecule, Vµ can be complicated compared to Vξ. Example: The water molecule For the water molecule, VRIG is two dimensional (as it is for any system with G = SO(3)). The definition gives, at a configuration (r, s), and angular momentum µ, VRIG = { (η × r,η × s) | η ∈ R3 satisfies (m 2 ‖r‖2 + 2Mm‖s‖2 ) η · µ − m 2 (r · η)(r · µ) + 2Mm(s · µ)(s · η) = 0 }. The condition on η is just the condition η ∈ g⊥µ . The internal space is three dimensional, the dimension of shape space. The definition gives VINT = { (δr, δs) | (η · ξ)(mr · δr + 4Mms · δs) − m 2 [(δr · ξ)(r · η) + (r · ξ)(δr · η)] − 2Mm[(δs · ξ)(s · η) + (s · ξ)(δs · η)] = 0 for all ξ,η ∈ g⊥µ }. 5.4 The Normal Form for the Symplectic Structure One of the most interesting aspects of block diagonalization is that the rigid-internal splitting introduced in the last section also brings the sym- plectic structure into normal form. We already gave the general structure of 5.4 The Normal Form for the Symplectic Structure 115 this and here we provide a few more details. We emphasize once more that this implies that the equations of motion are also put into normal form and this is useful for studying eigenvalue movement for purposes of bifurcation theory. For example, for Abelian groups, the linearized equations of motion take the gyroscopic form: Mq¨ + Sq˙ + Λq = 0 where M is a positive definite symmetric matrix (the mass matrix), Λ is symmetric (the potential term) and S is skew (the gyroscopic, or magnetic term). This second order form is particularly useful for finding eigenval- ues of the linearized equations (see, for example, Bloch, Krishnaprasad, Marsden, and Ratiu [1991, 1994]). To make the normal form of the symplectic structure explicit, we shall need a preliminary result. 5.4.1 Lemma. Let ∆q = ηQ(qe) ∈ VRIG and ∆z = TAµ · ∆q ∈ SRIG. Then ∆z = vert [FL(ζQ(qe))]− T ∗ηQ(qe) · pe (5.4.1) where ζ = I(qe)−1ad∗ηµ and vert denotes the vertical lift. Proof. We shall give a coordinate proof and leave it to the reader to supply an intrinsic one. In coordinates, Aµ is given by (3.3.8) as (Aµ)i = gijK j bµaIab. Differentiating, [TAµ · ν]i = ∂gij ∂qk νkKjbµaI ab + gij ∂Kjb ∂qk νkµaIab + gijKjbµa ∂Iab ∂qk νk. (5.4.2) The fact that the action consists of isometries gives an identity allowing us to eliminate derivatives of gij : ∂gij ∂qk Kkaη a + gkj ∂Kka ∂qi ηa + gik ∂Kka ∂qj ηa = 0. (5.4.3) Substituting this in (5.4.2), taking νm = Kma η a, and rearranging terms gives (∆z)i = { gik ∂Kkb ∂qm Kma − gik ∂Kka ∂qm Kmb − gkj ∂Kka ∂qi Kjb } ηaµcIbc + gijK j bK k c ∂Iab ∂qk µaη c. (5.4.4) For group actions, one has the general identity [ηQ, ζQ] = −[η, ζ]Q which gives ∂Kka ∂qm Kmb − ∂Kkb ∂qm Kma = K k cC c ab. (5.4.5) 116 5. The Energy–Momentum Method This allows one to simplify the first two terms in (5.4.4) giving (∆z)i = −gikKkdCdabηaµcIbc − gkj ∂Kka ∂qi Kjbη aµcIbc + gijKjbK k c ∂Iab ∂qk µaη c. (5.4.6) Next, employ the identity (5.3.14): ∂Iab ∂qk Kkcη c = Caedη eIdb + IaeCbdeηd. (5.4.7) Substituting (5.4.7) into (5.4.6), one pair of terms cancels, leaving (∆z)i = gikKkdC c abη aµcIbd − gkj ∂K k a ∂qi Kjbη aµcIbc (5.4.8) which is exactly (5.4.1) since at a relative equilibrium, p = Aµ(q); that is, pk = gkjK j bIbcµc. Now we can start computing the items in the symplectic form. 5.4.2 Lemma. For any δz ∈ TzeP , Ω(ze)(∆z, δz) = 〈[dJ(ze) · δz], η〉 − 〈〈ζQ(qe), δq〉〉. (5.4.9) Proof. Again, a coordinate calculation is convenient. The symplectic form on (∆p,∆q), (δp, δq) is Ω(ze)(∆z, δz) = (δp)i(∆q)i − (∆p)i(δq)i = δpiKiaηa − (∆p)i(δqi). Substituting from (5.4.8), and using 〈(dJ · δz), η〉 = δpiKiaηa + pi ∂Kia ∂qk δqkηa gives (5.4.9). If δz ∈ SINT, then it lies in ker dJ, so we get the internal-rigid inter- action terms: Ω(ze)(∆z, δz) = −〈〈ζQ(qe), δq〉〉. (5.4.10) Since these involve only δq and not δp, there is a zero in the last slot in the first row of Ω. 5.4.3 Lemma. The rigid-rigid terms in Ω are Ω(ze)(∆1z,∆2z) = −〈µ, [η1, η2]〉, (5.4.11) which is the coadjoint orbit symplectic structure. 5.5 Stability of Relative Equilibria for the Double Spherical Pendulum 117 Proof. By Lemma 5.4.9, with ζ1 = I−1ad∗η1µ, Ω(ze)(∆1z,∆2z) = −〈〈ζ1Q(qe), η2Q(qe)〉〉 = −〈ad∗η1µ, η2〉 = −〈µ, adη1η2〉. Next, we turn to the magnetic terms: 5.4.4 Lemma. Let δ1z = TAµ · δ1q and δ2z = TAµ · δ2q ∈ WINT, where δ1q, δ2q ∈ VINT. Then Ω(ze)(δ1z, δ2z) = −dAµ(δ1q, δ2q). (5.4.12) Proof. Regarding Aµ as a map of Q to T ∗Q, and recalling that ze = Aµ(qe), we recognize the left hand side of (5.4.12) as the pull back of Ω by αµ (and then restricted to VINT). However, as we saw already in Chapter 2, the canonical one-form is characterized by β∗Θ = β, so β∗Ω = −dβ for any one-form β. Therefore, Ω(ze)(δ1z, δ2z) = −dAµ(qe)(δ1q, δ2q). If we define the one form Aξ by Aξ(q) = FL(ξQ(q)), then the definition of VINT shows that on this space dAµ = dAξ. This is a useful remark since dAξ is somewhat easier to compute in examples. If we had made the “naive” choice of VINT as the orthogonal complement of the G-orbit, then we could also replace dAµ by 〈µ, curvA〉. However, with our choice of VINT, one must be careful of the distinction. We leave it for the reader to check that the rest of theWINT,W∗INT block is as stated. 5.5 Stability of Relative Equilibria for the Double Spherical Pendulum We now give some of the results for the stability of the branches of the double spherical pendulum that we found in the last chapter. We refer the reader to Marsden and Scheurle [1993a] for additional details. Even though the symmetry group of this example is Abelian, and so there is no rigid body block, the calculations are by no means trivial. We shall leave the simple pendulum to the reader, in which case all of the relative equilibria are stable, except for the straight upright solution, which is unstable. The water molecule is a nice illustration of the general structure of the method. We shall not work this example out here, however, as it is quite complicated, as we have mentioned. However, we shall come back to it in Chapter 8 and indicate some more general structure that can be obtained on grounds of discrete symmetry alone. For other informative examples, the reader can consult Lewis and Simo [1990], Simo, Lewis, and Marsden 118 5. The Energy–Momentum Method [1991], Simo, Posbergh, and Marsden [1990, 1991], and Zombro and Holmes [1993]. To carry out the stability analysis for relative equilibria of the double spherical pendulum, one must compute δ2Vµ on the subspace orthogonal to the Gµ-orbit. To do this, is is useful to introduce coordinates adapted to the problem and to work in Lagrangian representation. Specifically, let q⊥1 and q⊥2 be given polar coordinates (r1, θ1) and (r2, θ2) respectively. Then ϕ = θ2 − θ1 represents an S1-invariant coordinate, the angle between the two vertical planes formed by the pendula. In these terms, one computes from our earlier expressions that the angular momentum is J = (m1 +m2)r21 θ˙1 +m2r 2 2 θ˙2 +m2r1r2(θ˙1 + θ˙2) cosϕ+m2(r1r˙2 − r2r˙1) sinϕ (5.5.1) and the Lagrangian is L = 1 2 m1(r˙21 + r 2 1 θ˙ 2 1) + 1 2 m2 { r˙21 + r 2 1 θ˙ 2 1 + r˙ 2 2 + r 2 2 θ˙ 2 2 +2(r˙1r˙2 + r1r2θ˙1θ˙2) cosϕ+ 2(r1r˙2θ˙1 − r2r˙1θ˙2) sinϕ } + 1 2 m1 r21 r˙ 2 1 l21 − r21 + 1 2 m2 ( r1r˙1√ l21 − r21 + r2r˙2√ l22 − r22 )2 +m1g √ l21 − r21 +m2g (√ l21 − r21 + √ l22 − r22 ) . (5.5.2) One also has, from (3.5.17), Vµ = −m1g √ l21 − r21 −m2g (√ l21 − r21 + √ l22 − r22 ) + 1 2 µ2 m1r21 +m2(r 2 1 + r 2 2 + 2r1r2 cosϕ) . (5.5.3) Notice that Vµ depends on the angles θ1 and θ2 only through ϕ = θ2−θ1, as it should by S1-invariance. Next one calculates the second variation at one of the relative equilibria found in §4.3. If we calculate it as a 3× 3 matrix in the variables r1, r2, ϕ, then one checks that we will automatically be in a space orthogonal to the Gµ-orbits. One finds, after some computation, that δ2Vµ =  a b 0b d 0 0 0 e  (5.5.4) 5.5 Stability of Relative Equilibria for the Double Spherical Pendulum 119 where a = µ2(3(m+ α)2 − α2(m− 1)) λ4l41m2(m+ α2 + 2α)3 + gm2m l1(1− λ2)3/2 , b = (signα) µ2 λ4l41m2 3(m+ α2 + 2α) + 4α(m− 1) (m+ α2 + 2α)3 , d = µ2 λ4l41m2 3(α+ 1)2 + 1−m (m+ α2 + 2α)3 + m2g l1 r2 (r2 − λ2α2)3/2 , e = µ2 λ2l21m2 α (m+ α2 + 2α)2 . Notice the zeros in (5.5.4); they are in fact a result of discrete symmetry, as we shall see in Chapter 8. Without the help of these zeros (for example, if the calculation is done in arbitrary coordinates), the expression for δ2Vµ might be intractable. Based on this calculation one finds: 5.5.1 Proposition. The signature of δ2Vµ along the “straight out” branch of the double spherical pendulum (with α > 0) is (+,+,+) and so is sta- ble. The signature along the “cowboy branch” with α < 0 and emanating from the straight down state (λ = 0) is (−,−,+) and along the remaining branches is (−,+,+). The stability along the cowboy branch requires further analysis that we shall outline in Chapter 10. The remaining branches are linearly unstable since the index is odd (this is for reasons we shall go into in Chapter 10). To get this instability and bifurcation information, one needs to linearize the reduced equations and compute the corresponding eigenvalues. There are (at least) three methodologies that can be used for computing the reduced linearized equations: (i) Compute the Euler–Lagrange equations from (5.5.2), drop them to J−1(µ)/Gµ and linearize the resulting equations. (ii) Read off the linearized reduced equations from the block diagonal form of δ2Hξ and the symplectic structure. (iii) Perform Lagrangian reduction as in §3.6 to obtain the Lagrangian structure of the reduced system and linearize it at a relative equilib- rium. For the double spherical pendulum, perhaps the first method is the quickest to get the answer, but of course the other methods provide insight and information about the structure of the system obtained. The linearized system obtained has the following standard form expected for Abelian reduction: Mq¨ + Sq˙ + Λq = 0. (5.5.5) 120 5. The Energy–Momentum Method In our case q = (r1, r2, ϕ) and Λ is the matrix (5.5.4) given above. The mass matrix M is M =  m11 m12 0m12 m22 0 0 0 m33  where m11 = m1 +m2 1− λ2 , m12 = (signα)m2 ( 1 + αλ2√ 1− λ2√r2 − α2λ2 ) , m22 = m2 r2 r2 − λ2α2 , m33 = m2l 2 1λ 2(m− 1) α 2 m+ α2 + 2α , and the gyroscopic matrix S (the magnetic term) is S =  0 0 s130 0 s23 −s13 −s23 0  where s13 = µ λl1 2α2(m− 1) (m+ α2 + 2α)2 and s23 = −(signα) µ λl1 2α(m− 1) (m+ α2 + 2α)2 . We will pick up this discussion again in Chapter 10. Page 121 6 Geometric Phases In this chapter we give the basic ideas for geometric phases in terms of reconstruction, prove Montgomery’s formula for rigid body phases, and give some other basic examples. We refer to Marsden, Montgomery, and Ratiu [1990] and to Marsden, Ratiu and Scheurle [2000] for more information. 6.1 A Simple Example In Chapter 1 we discussed the idea of phases and gave several examples in general terms. What follows is a specific but very simple example to illustrate the important role played by the conserved quantity (in this case the angular momentum). Consider two planar rigid bodies joined by a pin joint at their centers of mass. Let I1 and I2 be their moments of inertia and θ1 and θ2 be the angle they make with a fixed inertial direction, as in Figure 6.1.1. Conservation of angular momentum states that I1θ˙1 + I2θ˙2 = µ = con- stant in time, where the overdot means time derivative. Recall from Chapter 3 that the shape space Q/G of a system is the space whose points give the shape of the system. In this case, shape space is the circle S1 parametrized by the hinge angle ψ = θ2 − θ1. We can parametrize the configuration space of the system by θ1 and θ2 or by θ = θ1 and ψ. Conservation of angular momentum reads I1θ˙ + I2(θ˙ + ψ˙) = µ; that is, dθ + I2 I1 + I2 dψ = µ I1 + I2 dt. (6.1.1) 122 6. Geometric Phases inertial fram e body # 1 body # 2 Figure 6.1.1. Two rigid bodies coupled at their centers of mass. The left hand side of (6.1.1) is the mechanical connection discussed in Chapter 3, where ψ parametrizes shape space Q/G and θ parametrizes the fiber of the bundle Q→ Q/G. Suppose that body #2 goes through one full revolution so that ψ in- creases from 0 to 2pi. Suppose, moreover, that the total angular momentum is zero: µ = 0. From (6.1.1) we see that ∆θ = − I2 I1 + I2 ∫ 2pi 0 dψ = − ( I2 I1 + I2 ) 2pi. (6.1.2) This is the amount by which body #1 rotates, each time body #2 goes around once. This result is independent of the detailed dynamics and only depends on the fact that angular momentum is conserved and that body #2 goes around once. In particular, we get the same answer even if there is a “hinge potential” hindering the motion or if there is a control present in the joint. Also note that if we want to rotate body #1 by −2pikI2/(I1 + I2) radians, where k is an integer, all one needs to do is spin body #2 around k times, then stop it. By conservation of angular momentum, body #1 will stay in that orientation after stopping body #2. In particular, if we think of body #1 as a spacecraft and body #2 as an internal rotor, this shows that by manipulating the rotor, we have control over the attitude (orientation) of the spacecraft. Here is a geometric interpretation of this calculation. Define the one form A = dθ + I2 I1 + I2 dψ. (6.1.3) This is a flat connection for the trivial principal S1-bundle pi : S1×S1 → S1 given by pi(θ, ψ) = ψ. Formula (6.1.2) is the holonomy of this connection, when we traverse the base circle, 0 ≤ ψ ≤ 2pi. 6.2 Reconstruction 123 Another interesting context in which geometric phases comes up is the phase shift that occurs in interacting solitons. In fact, Alber and Marsden [1992] have shown how this is a geometric phase in the sense described in this chapter. In the introduction we already discussed a variety of other examples. 6.2 Reconstruction This section presents a reconstruction method for the dynamics of a given Hamiltonian system from that of the reduced system. Let P be a symplectic manifold on which a Lie group acts in a Hamiltonian manner and has a momentum map J : P → g∗. Assume that an integral curve cµ(t) of the reduced Hamiltonian vector field XHµ on the reduced space Pµ is known. For z0 ∈ J−1(µ), we search for the corresponding integral curve c(t) = Ft(z0) of XH such that piµ(c(t)) = cµ(t), where piµ : J−1(µ) → Pµ is the projection. To do this, choose a smooth curve d(t) in J−1(µ) such that d(0) = z0 and piµ(d(t)) = cµ(t). Write c(t) = Φg(t)(d(t)) for some curve g(t) in Gµ to be determined, where the group action is denoted g · z = Φg(z). First note that XH(c(t)) = c′(t) = Td(t)Φg(t)(d′(t)) + Td(t)Φg(t) · ( Tg(t)Lg(t)−1(g′(t)) ) P (d(t)). (6.2.1) Since Φ∗gXH = XΦ∗gH = XH , (6.2.1) gives d′(t) + ( Tg(t)Lg(t)−1(g′)(t) ) P (d(t)) = TΦg(t)−1XH(Φg(t)(d(t))) = (Φ∗g(t)XH)(d(t)) = XH(d(t)). (6.2.2) This is an equation for g(t) written in terms of d(t) only. We solve it in two steps: Step 1. Find ξ(t) ∈ gµ such that ξ(t)P (d(t)) = XH(d(t))− d′(t). (6.2.3) Step 2. With ξ(t) determined, solve the following non-autonomous ordi- nary differential equation on Gµ: g′(t) = TeLg(t)(ξ(t)), with g(0) = e. (6.2.4) Step 1 is typically of an algebraic nature; in coordinates, for matrix Lie groups, (6.2.3) is a matrix equation. We show later how ξ(t) can be 124 6. Geometric Phases explicitly computed if a connection is given on J−1(µ) → Pµ. With g(t) determined, the desired integral curve c(t) is given by c(t) = Φg(t)(d(t)). A similar construction works on P/G, even if the G-action does not admit a momentum map. Step 2 can be carried out explicitly when G is Abelian. Here the con- nected component of the identity of G is a cylinder Rp ×Tk−p and the ex- ponential map exp(ξ1, . . . , ξk) = (ξ1, . . . , ξp, ξp+1(mod2pi), . . . , ξk(mod2pi)) is onto, so we can write g(t) = exp η(t), where η(0) = 0. Therefore ξ(t) = Tg(t)Lg(t)−1(g′(t)) = η′(t) since η′ and η commute, that is, η(t) = ∫ t 0 ξ(s)ds. Thus the solution of (6.2.4) is g(t) = exp (∫ t 0 ξ(s)ds ) . (6.2.5) This reconstruction method depends on the choice of d(t). With addi- tional structure, d(t) can be chosen in a natural geometric way. What is needed is a way of lifting curves on the base of a principal bundle to curves in the total space, which can be done using connections. One can object at this point, noting that reconstruction involves integrating one ordinary differential equation, whereas introducing a connection will involve integra- tion of two ordinary differential equations, one for the horizontal lift and one for constructing the solution of (6.2.4) from it. However, for the deter- mination of phases, there are some situations in which the phase can be computed without actually solving either equation, so one actually solves no differential equations; a specific case is the rigid body. Suppose that piµ : J−1(µ) → Pµ is a principal Gµ-bundle with a con- nection A. This means that A is a gµ-valued one-form on J−1(µ) ⊂ P satisfying (i) Ap · ξP (p) = ξ for ξ ∈ gµ, (ii) L∗gA = Adg ◦A for g ∈ Gµ. Let d(t) be the horizontal lift of cµ through z0, that is, d(t) satisfies A(d(t)) · d′(t) = 0, piµ ◦ d = cµ, and d(0) = z0. We summarize: 6.2.1 Theorem (Reconstruction). Suppose piµ : J−1(µ)→ Pµ is a prin- cipal Gµ-bundle with a connection A. Let cµ be an integral curve of the reduced dynamical system on Pµ. Then the corresponding curve c through a point z0 ∈ pi−1µ (cµ(0)) of the system on P is determined as follows: (i) Horizontally lift cµ to form the curve d in J−1(µ) through z0. (ii) Let ξ(t) = A(d(t)) ·XH(d(t)), so that ξ(t) is a curve in gµ. (iii) Solve the equation g˙(t) = g(t) · ξ(t) with g(0) = e. Then c(t) = g(t) · d(t) is the integral curve of the system on P with initial condition z0. 6.3 Cotangent Bundle Phases—a Special Case 125 Suppose cµ is a closed curve with period T ; thus, both c and d reintersect the same fiber. Write d(T ) = gˆ · d(0) and c(T ) = h · c(0) for gˆ, h ∈ Gµ. Note that h = g(T )gˆ. (6.2.6) The Lie group element gˆ (or the Lie algebra element log gˆ) is called the geometric phase . It is the holonomy of the path cµ with respect to the connection A and has the important property of being parametrization in- dependent. The Lie group element g(T ) (or log g(T )) is called the dynamic phase . For compact or semi-simple G, the group Gµ is generically Abelian. The computation of g(T ) and gˆ are then relatively easy, as was indicated above. 6.3 Cotangent Bundle Phases—a Special Case We now discuss the case in which P = T ∗Q, and G acts on Q and therefore on P by cotangent lift. In this case the momentum map is given by the formula J(αq) · ξ = αq · ξQ(q) = ξT∗Q(αq) y θ(αq), where ξ ∈ gµ, αq ∈ T ∗qQ, θ = pidqi is the canonical one-form and y is the interior product. Assume Gµ is the circle group or the line. Pick a generator ζ ∈ gµ, ζ 6= 0. For instance, one can choose the shortest ζ such that exp(2piζ) = 1. Identify gµ with the real line via ω 7→ ωζ. Then a connection one-form is a standard one-form on J−1(µ). 6.3.1 Proposition. Suppose Gµ ∼= S1 or R. Identify gµ with R via a choice of generator ζ. Let θµ denote the pull-back of the canonical one- form to J−1(µ). Then A = 1 〈µ, ζ〉θµ ⊗ ζ is a connection one-form on J−1(µ)→ Pµ. Its curvature as a two-form on the base Pµ is Ω = − 1〈µ, ζ〉ωµ, where ωµ is the reduced symplectic form on Pµ. Proof. Since G acts by cotangent lift, it preserves θ, and so θµ is pre- served by Gµ and therefore A is Gµ-invariant. Also, A ·ζP = [ζP y θ/〈µ, ζ〉]ζ = [〈J, ζ〉/〈µ, ζ〉]ζ = ζ. This verifies that A is a connection. The calculation of its curvature is straightforward. 126 6. Geometric Phases Remarks. 1. The result of Proposition 6.3.1 holds for any exact symplectic mani- fold. We shall use this for the rigid body. 2. In the next section we shall show how to construct a connection on J−1(µ) → Pµ in general. For the rigid body it is easy to check that these two constructions coincide! If G = Q and Gµ = S1, then the construction in Proposition 6.3.1 agrees with the pullback of the me- chanical connection for G semisimple and the metric defined by the Cartan-Killing form. 3. Choose P to be a complex Hilbert space with symplectic form Ω(ϕ,ψ) = −Im〈ϕ,ψ〉 and S1 action given by eiθψ. The corresponding momentum map is J(ϕ) = −‖ϕ‖2/2. Write Ω = −dΘ where Θ(ϕ) · ϕ = 12 Im〈ϕ,ψ〉. Now we identify the reduced space at level −1/2 to get projective Hilbert space. Applying Proposition 6.3.1, we get the basic phase result for quantum mechanics due to Aharonov and Anandan [1987]: The holonomy of a loop in projective complex Hilbert space is the exponential of twice the symplectic area of any two dimensional submanifold whose boundary is the given loop. 6.4 Cotangent Bundles—General Case If Gµ is not Abelian, the formula for A given above does not satisfy the second axiom of a connection. However, if the bundle Q → Q/Gµ has a connection, we will show below how this induces a connection on J−1(µ)→ (T ∗Q)µ. To do this, we can use the cotangent bundle reduction theorem. Recall that the mechanical connection provides a connection on Q→ Q/G and also on Q→ Q/Gµ. Denote the latter connection by γ. Denote by Jµ : T ∗Q→ g∗µ the induced momentum map, that is, Jµ(αq) = J(αq)|gµ. From the cotangent bundle reduction theorem, it follows that the diagram in Figure 6.4.1 commutes. In this figure, tµ(αq) = αq − µ′ · γq(·) is fiber translation by the µ′- component of the connection form and where [αq − µ′ · γq(·)] means the element of T ∗(Q/Gµ) determined by αq −µ′ · γq(·) and µ′ = µ|gµ. Call the composition of the two maps on the bottom of this diagram σ : [αq] ∈ (T ∗Q)µ 7→ [q] ∈ Q/Gµ. We induce a connection on J−1(µ)→ (T ∗Q)µ by being consistent with this diagram. 6.4 Cotangent Bundles—General Case 127 J−1(µ) - J−1µ (µ ′) - tµ J−1µ (0) - i T ∗Q - pi Q (T ∗Q)µ - T ∗(Q/Gµ) - Q/Gµ ? piµ /Gµ ? ρµ ? [αq] - [αq − µ′ · γd(·)] - [q] Figure 6.4.1. Notation for cotangent bundles. 6.4.1 Proposition. The connection one-form γ induces a connection one-form γ˜ on J−1(µ) by pull-back: γ˜ = (pi ◦ tµ)∗γ, that is, γ˜(αq) · Uαq = γ(q) · Tαqpi(Uαq ), for αq ∈ T ∗qQ, Uαq ∈ Tαq (T ∗Q). Similarly curv(γ˜) = (pi ◦ tµ)∗curv(γ) and in particular the µ′-component of the curvature of this connection is the pull-back of µ′ · curv(γ). The proof is a direct verification. 6.4.2 Proposition. Assume that ρµ : Q → Q/Gµ is a principal Gµ- bundle with a connection γ. If H is a G-invariant Hamiltonian on T ∗Q inducing the Hamiltonian Hµ on (T ∗Q)µ and cµ(t) is an integral curve of XHµ denote by d(t) a horizontal lift of cµ(t) in J −1(µ) relative to the natural connection of Proposition 6.4.1 and let q(t) = pi(d(t)) be the base integral curve of c(t). Then ξ(t) of step (ii) in Theorem 6.2.1 is given by ξ(t) = γ(q(t)) · FH(d(t)), where FH : T ∗Q→ TQ is the fiber derivative of H, that is, FH(αq) · βq = d dtH(αq + tβq) ∣∣ t=0 . Proof. γ˜ ·XH = γ · Tpi ·XH = γ · FH. 6.4.3 Proposition. With γ the mechanical connection, step ii in Theo- rem 6.2.1 is equivalent to (ii′) ξ(t) ∈ gµ is given by ξ(t) = γ(q(t)) · d(t)], where d(t) ∈ T ∗q(t)Q. Proof. Apply Proposition 6.4.1 and use the fact that FH(αq) = a]q. (In coordinates, ∂H/∂pµ = gµνpν .) To see that the connection of Proposition 6.4.1 coincides with the one in §6.3 for the rigid body, use the fact that Q = G and αµ must lie in J−1(µ), so αµ is the right invariant one form equalling µ at g = e and that Gµ = S1. 128 6. Geometric Phases 6.5 Rigid Body Phases We now derive a formula of Goodman and Robinson [1958] and Mont- gomery [1991] for the rigid body phase (see also Marsden, Montgomery, and Ratiu [1990]). (See Marsden and Ratiu [1999] for more historical in- formation.) As we have seen, the motion of a rigid body is a geodesic with respect to a left-invariant Riemannian metric (the inertia tensor) on SO(3). The corresponding phase space is P = T ∗ SO(3) and the momentum map J : P → R3 for the left SO(3) action is right translation to the identity. We identify so(3)∗ with so(3) via the Killing form and identify R3 with so(3) via the map v 7→ vˆ where vˆ(w) = v×w, and where × is the standard cross product. Points in so(3)∗ are regarded as the left reduction of points in T ∗ SO(3) by SO(3) and are the angular momenta as seen from a body-fixed frame. The reduced spaces J−1(µ)/Gµ are identified with spheres in R3 of Euclidean radius ‖µ‖, with their symplectic form ωµ = −dS/‖µ‖ where dS is the standard area form on a sphere of radius ‖µ‖ and where Gµ consists of rotations about the µ-axis. The trajectories of the reduced dynamics are obtained by intersecting a family of homothetic ellipsoids (the energy ellipsoids) with the angular momentum spheres. In particular, all but four of the reduced trajectories are periodic. These four exceptional trajectories are the homoclinic trajectories. Suppose a reduced trajectory Π(t) is given on Pµ, with period T . After time T , by how much has the rigid body rotated in space? The spatial angular momentum is pi = µ = gΠ, which is the conserved value of J. Here g ∈ SO(3) is the attitude of the rigid body and Π is the body angular momentum. If Π(0) = Π(T ) then µ = g(0)Π(0) = g(T )Π(T ) and so g(T )−1µ = g(0)−1µ that is, g(T )g(0)−1 is a rotation about the axis µ. We want to compute the angle of this rotation. To answer this question, let c(t) be the corresponding trajectory in J−1(µ) ⊂ P . Identify T ∗ SO(3) with SO(3)×R3 by left trivialization, so c(t) gets identified with (g(t),Π(t)). Since the reduced trajectory Π(t) closes af- ter time T , we recover the fact that c(T ) = gc(0) where g = g(T )g(0)−1 ∈ Gµ. Thus, we can write g = exp[(∆θ)ζ] (6.5.1) where ζ = µ/‖µ‖ identifies gµ with R by aζ 7→ a, for a ∈ R. Let D be one of the two spherical caps on S2 enclosed by the reduced trajectory, Λ be 6.5 Rigid Body Phases 129 the corresponding oriented solid angle, with |Λ| = (areaD)/‖µ‖2, and let Hµ be the energy of the reduced trajectory. All norms are taken relative to the Euclidean metric of R3. We shall prove below that modulo 2pi, we have ∆θ = 1 ‖µ‖ {∫ D ωµ + 2HµT } = −Λ + 2HµT‖µ‖ . (6.5.2) The special case of this formula for a symmetric free rigid body was given by Hannay [1985] and Anandan [1988]. Other special cases were given by Ishlinskii [1952]. To prove (6.5.2), we choose the connection one-form on J−1(µ) to be the one from Proposition 6.3.1, or equivalently from Proposition 6.4.1: A = 1 ‖µ‖θµ, (6.5.3) where θµ is the pull back to J−1(µ) of the canonical one-form θ on T ∗ SO(3). The curvature of A as a two-form on the base Pµ, the sphere of radius ‖µ‖ in R3, is given by − 1‖µ‖ωµ = 1 ‖µ‖2 dS. (6.5.4) The first terms in (6.5.2) represent the geometric phase, that is, the holon- omy of the reduced trajectory with respect to this connection. The loga- rithm of the holonomy (modulo 2pi) is given as minus the integral over D of the curvature, that is, it equals 1 ‖µ‖ ∫ D ωµ = − 1‖µ‖2 (areaD) = −Λ(mod 2pi). (6.5.5) The second terms in (6.5.2) represent the dynamic phase. By Theorem 6.3.1, it is calculated in the following way. First one horizontally lifts the reduced closed trajectory Π(t) to J−1(µ) relative to the connection (6.5.3). This horizontal lift is easily seen to be (identity, Π(t)) in the left trivializa- tion of T ∗ SO(3) as SO(3)× R3. Second, we need to compute ξ(t) = (A ·XH)(Π(t)). (6.5.6) Since in coordinates θµ = pidqi and XH = pi ∂ ∂qi + ∂ ∂p terms for pi = ∑ j g ijpj , where gij is the inverse of the Riemannian metric gij on SO(3), we get (θµ ·XH)(Π(t)) = pipi = 2H(identity,Π(t)) = 2Hµ, (6.5.7) 130 6. Geometric Phases where Hµ is the value of the energy on S2 along the integral curve Π(t). Consequently, ξ(t) = 2Hµ ‖µ‖ ζ. (6.5.8) Third, since ξ(t) is independent of t, the solution of the equation g˙ = gξ = 2Hµ ‖µ‖ gζ is g(t) = exp ( 2Hµt ‖µ‖ ζ ) so that the dynamic phase equals ∆θd = 2Hµ ‖µ‖ T (mod 2pi). (6.5.9) Formulas (6.5.5) and (6.5.9) prove (6.5.2). Note that (6.5.2) is independent of which spherical cap one chooses amongst the two bounded by Π(t). Indeed, the solid angles on the unit sphere defined by the two caps add to 4pi, which does not change Formula (6.5.2). For other examples of the use of (6.5.2) see Chapter 7 and Levi [1993]. We also note that Goodman and Robinson [1958] and Levi [1993] give an interesting link between this result, the Poinsot description of rigid body motion and the Gauss–Bonnet theorem. 6.6 Moving Systems The techniques above can be merged with those for adiabatic systems, with slowly varying parameters. We illustrate the ideas with the example of the bead in the hoop discussed in Chapter 1. Begin with a reference configuration Q and a Riemannian manifold S. Let M be a space of embeddings of Q into S and let mt be a curve in M . If a particle in Q is following a curve q(t), and if we let the configuration space Q have a superimposed motion mt, then the path of the particle in S is given by mt(q(t)). Thus, its velocity in S is given by the time derivative: Tq(t)mt · q˙(t) + Zt(mt(q(t))) (6.6.1) where Zt, defined by Zt(mt(q)) = ddtmt(q), is the time dependent vector field (on S with domain mt(Q)) generated by the motion mt and Tq(t)mt ·w is the derivative (tangent) of the map mt at the point q(t) in the direction w. To simplify the notation, we write mt = Tq(t)mt and q(t) = mt(q(t)). Consider a Lagrangian on TQ of the form kinetic minus potential energy. Using (6.6.1), we thus choose Lmt(q, v) = 1 2 ‖mt · v + Zt(q(t))‖2 − V (q)− U(q(t)) (6.6.2) 6.6 Moving Systems 131 where V is a given potential on Q and U is a given potential on S. Put on Q the (possibly time dependent) metric induced by the mapping mt. In other words, choose the metric on Q that makes mt into an isometry for each t. In many examples of mechanical systems, such as the bead in the hoop given below, mt is already a restriction of an isometry to a submanifold of S, so the metric on Q in this case is in fact time independent. Now we take the Legendre transform of (6.6.2), to get a Hamiltonian system on T ∗Q. Taking the derivative of (6.6.2) with respect to v in the direction of w gives: p · w = 〈mt · v + Zt(q(t)),mt · w〉q(t) = 〈mt · v + Zt(q(t))T ,mt · w〉q(t) where p · w means the natural pairing between the covector p ∈ T ∗q(t)Q and the vector w ∈ Tq(t)Q, 〈 , 〉q(t) denotes the metric inner product on the space S at the point q(t) and T denotes the tangential projection to the space mt(Q) at the point q(t). Recalling that the metric on Q, denoted 〈 , 〉q(t) is obtained by declaring mt to be an isometry, the above gives p · w = 〈v + m−1t Zt(q(t))T , w〉q(t); that is, p = (v + m−1t Zt(q(t))T )b (6.6.3) where b denotes the index lowering operation at q(t) using the metric on Q. The (in general time dependent) Hamiltonian is given by the prescription H = p · v − L, which in this case becomes Hmt(q, p) = 1 2 ‖p‖2 − P(Zt)− 12Z ⊥ t ‖2 + V (q) + U(q(t)) = H0(q, p)− P(Zt)− 12‖Z ⊥ t ‖2 + U(q(t)), (6.6.4) where H0(q, p) = 12‖p‖2 +V (q), the time dependent vector field Zt ∈ X(Q) is defined by Zt(q) = m−1t [Zt(mt(q))]T , the momentum function P(Y ) is defined by P(Y )(q, p) = p · Y (q) for Y ∈ X(Q), and where Z⊥t denotes the orthogonal projection of Zt to mt(Q). Even though the Lagrangian and Hamiltonian are time dependent, we recall that the Euler–Lagrange equa- tions for Lmt are still equivalent to Hamilton’s equations for Hmt . These give the correct equations of motion for this moving system. (An interesting example of this is fluid flow on the rotating earth, where it is important to consider the fluid with the motion of the earth superposed, rather than the motion relative to an observer. This point of view is developed in Chern [1991].) Let G be a Lie group that acts on Q. (For the bead in the hoop, this will be the dynamics of H0 itself.) We assume for the general theory that H0 is G-invariant. Assuming the “averaging principle” (cf. Arnold [1978], for example) we replace Hmt by its G-average, 〈Hmt〉(q, p) = 1 2 ‖p‖2 − 〈P(Zt)〉 − 12 〈‖Z ⊥ t ‖2〉+ V (q) + 〈U(q(t))〉 (6.6.5) 132 6. Geometric Phases where 〈 · 〉 denotes the G-average. This principle can be hard to rigorously justify in general. We will use it in a particularly simple example where we will see how to check it directly. Furthermore, we shall discard the term 1 2 〈‖Z⊥t ‖2〉; we assume it is small compared to the rest of the terms. Thus, define H(q, p, t) = 1 2 ‖p‖2 − 〈P(Zt)〉+ V (q) + 〈U(q(t))〉 = H0(q, p)− 〈P(Zt)〉+ 〈U(q(t))〉. (6.6.6) The dynamics of H on the extended space T ∗Q×M is given by the vector field (XH, Zt) = ( XH0 −X〈P(Zt)〉 +X〈U◦mt〉, Zt ) . (6.6.7) The vector field hor(Zt) = (−X〈P(Zt)〉, Zt) (6.6.8) has a natural interpretation as the horizontal lift of Zt relative to a con- nection, which we shall call the Hannay–Berry connection induced by the Cartan connection . The holonomy of this connection is interpreted as the Hannay–Berry phase of a slowly moving constrained system. 6.7 The Bead on the Rotating Hoop Consider Figures 1.5.2 and 1.5.3 which show a hoop (not necessarily circu- lar) on which a bead slides without friction. As the bead is sliding, the hoop is slowly rotated in its plane through an angle θ(t) and angular velocity ω(t) = θ˙(t)k. Let s denote the arc length along the hoop, measured from a reference point on the hoop and let q(s) be the vector from the origin to the corresponding point on the hoop; thus the shape of the hoop is determined by this function q(s). Let L be the length of the hoop. The unit tangent vector is q′(s) and the position of the reference point q(s(t)) relative to an inertial frame in space is Rθ(t)q(s(t)), where Rθ is the rotation in the plane of the hoop through an angle θ. The configuration space is diffeomorphic to the circle Q = S1. The La- grangian L(s, s˙, t) is the kinetic energy of the particle; that is, since d dt Rθ(t)q(s(t)) = Rθ(t)q′(s(t))s˙(t) +Rθ(t)[ω(t)× q(s(t))], we set L(s, s˙, t) = 1 2 m‖q′(s)s˙+ ω × q(s)‖2. (6.7.1) The Euler–Lagrange equations d dt ∂L ∂s˙ = ∂L ∂s 6.7 The Bead on the Rotating Hoop 133 become d dt m[s˙+ q′ · (ω× q)] = m[s˙q′′ · (ω× q) + s˙q′ · (ω× q′) + (ω× q) · (ω× q′)] since ‖q′‖2 = 1. Therefore s¨+ q′′ · (ω × q)s˙+ q′ · (ω˙ × q) = s˙q′′ · (ω × q) + (ω × q) · (ω × q′); that is, s¨− (ω × q) · (ω × q′) + q′ · (ω˙ × q) = 0. (6.7.2) The second and third terms in (6.7.2) are the centrifugal and Euler forces respectively. We rewrite (6.7.2) as s¨ = ω2q · q′ − ω˙q sinα (6.7.3) where α is as in Figure 1.5.2 and q = ‖q‖. From (6.7.3), Taylor’s formula with remainder gives s(t) = s0 + s˙0t (6.7.4) + ∫ t 0 (t− t′){ω(t′)2q · q′(s(t′))− ω˙(t′)q(s(t′)) sinα(s(t′))} dt′. Now ω and ω˙ are assumed small with respect to the particle’s velocity, so by the averaging theorem (see, e.g., Hale [1969]), the s-dependent quantities in (6.7.4) can be replaced by their averages around the hoop: s(T ) ≈ s0 + s˙0T (6.7.5) + ∫ T 0 (T − t′) { ω(t′)2 1 L ∫ L 0 q · q′ds− ω˙(t′) 1 L ∫ L 0 q(s) sinαds } dt′. Aside The essence of the averaging can be seen as follows. Suppose g(t) is a rapidly varying function and f(t) is slowly varying on an interval [a, b]. Over one period of g, say [α, β], we have∫ β α f(t)g(t)dt ≈ ∫ β α f(t)g¯dt (6.7.6) where g¯ = 1 β − α ∫ β α g(t)dt is the average of g. The error in (6.7.6) is∫ β α f(t)(g(t)− g¯)dt 134 6. Geometric Phases which is less than (β − α)× (variation of f)× constant ≤ constant ‖f ′‖(β − α)2. If this is added up over [a, b] one still gets something small as the period of g tends to zero. The first integral in (6.7.5) over s vanishes and the second is 2A where A is the area enclosed by the hoop. Now integrate by parts:∫ T 0 (T − t′)ω˙(t′)dt′ = −Tω(0) + ∫ T 0 ω(t′)dt′ = −Tω(0) + 2pi, (6.7.7) assuming the hoop makes one complete revolution in time T . Substitut- ing (6.7.7) in (6.7.5) gives s(T ) ≈ s0 + s˙0T + 2A L ω0T − 4piA L . (6.7.8) The initial velocity of the bead relative to the hoop is s˙0, while that relative to the inertial frame is (see (6.7.1)), v0 = q′(0) · [q′(0)s˙0 + ω0 × q(0)] = s˙0 + ω0q(s0) sinα(s0). (6.7.9) Now average (6.7.8) and (6.7.9) over the initial conditions to get 〈s(T )− s0 − v0T 〉 ≈ −4piA L (6.7.10) which means that on average, the shift in position is by 4piA/L between the rotated and nonrotated hoop. This extra length 4piA/L, (or in angular measure, 8pi2A/L2) is the Hannay–Berry phase. Note that if ω0 = 0 (the situation assumed by Berry [1985]) then averaging over initial conditions is not necessary. This process of averaging over the initial conditions used in this example is related to work of Golin and Marmi [1990] on procedures to measure the phase shift. Page 135 7 Stabilization and Control In this chapter we present a result of Bloch, Krishnaprasad, Marsden, and Sanchez de Alvarez [1992] on the stabilization of rigid body motion using internal rotors followed by a description of some of Montgomery [1990] work on optimal control. Some other related results will be presented as well. 7.1 The Rigid Body with Internal Rotors Consider a rigid body (to be called the carrier body) carrying one, two or three symmetric rotors. Denote the system center of mass by 0 in the body frame and at 0 place a set of (orthonormal) body axes. Assume that the rotor and the body coordinate axes are aligned with principal axes of the carrier body. The configuration space of the system is SO(3)×S1×S1×S1. Let Ibody be the inertia tensor of the carrier, body, Irotor the diagonal matrix of rotor inertias about the principal axes and I′rotor the remaining rotor inertias about the other axes. Let Ilock = Ibody + Irotor + I′rotor be the (body) locked inertia tensor (i.e., with rotors locked) of the full system; this definition coincides with the usage in Chapter 3, except that here the locked inertia tensor is with respect to body coordinates of the carrier body (note that in Chapter 3, the locked inertia tensor is with respect to the spatial frame). 136 7. Stabilization and Control The Lagrangian of the free system is the total kinetic energy of the body plus the total kinetic energy of the rotor; that is, L = 1 2 (Ω · IbodyΩ) + 12Ω · I ′ rotorΩ + 1 2 (Ω + Ωr) · Irotor(Ω + Ωr) = 1 2 (Ω · (Ilock − Irotor)Ω) + 12(Ω + Ωr) · Irotor(Ω + Ωr) (7.1.1) where Ω is the vector of body angular velocities and Ωr is the vector of rotor angular velocities about the principal axes with respect to a (carrier) body fixed frame. Using the Legendre transform, we find the conjugate momenta to be: m = ∂L ∂Ω = (Ilock − Irotor)Ω + Irotor(Ω + Ωr) = IlockΩ + IrotorΩr (7.1.2) and l = ∂L ∂Ωr = Irotor(Ω + Ωr) (7.1.3) and the equations of motion including internal torques (controls) u in the rotors are m˙ = m× Ω = m× (Ilock − Irotor)−1(m− l) l˙ = u. (7.1.4) 7.2 The Hamiltonian Structure with Feedback Controls The first result shows that for torques obeying a certain feedback law (i.e., the torques are given functions of the other variables), the preceding set of equations including the internal torques u can still be Hamiltonian! As we shall see, the Hamiltonian structure is of gyroscopic form. 7.2.1 Theorem. For the feedback u = k(m× (Ilock − Irotor)−1(m− l)), (7.2.1) where k is a constant real matrix such that k does not have 1 as an eigen- value and such that the matrix J = (1 − k)−1(Ilock − Irotor) is symmetric, the system (7.1.4) reduces to a Hamiltonian system on so(3)∗ with respect to the rigid body bracket {F,G}(m) = −m · (∇F ×∇G). Proof. We have l˙ = u = km˙ = k((IlockΩ + IrotorΩr)× Ω). (7.2.2) Therefore, the vector km− l = p, (7.2.3) 7.2 The Hamiltonian Structure with Feedback Controls 137 is a constant of motion. Hence our feedback control system becomes m˙ = m× (Ilock − Irotor)−1(m− l) = m× (Ilock − Irotor)−1(m− km+ p) = m× (Ilock − Irotor)−1(1− k)(m− ξ) (7.2.4) where ξ = −(1 − k)−1p and 1 is the identity. Define the k-dependent “inertia tensor” J = (1− k)−1(Ilock − Irotor). (7.2.5) Then the equations become m˙ = ∇C ×∇H (7.2.6) where C = 1 2 ‖m‖2 (7.2.7) and H = 1 2 (m− ξ) · J−1(m− ξ). (7.2.8) Clearly (7.2.6) are Hamiltonian on so(3)∗ with respect to the standard Lie–Poisson structure (see §2.6). The conservation law (7.2.3), which is key to our methods, may be re- garded as a way to choose the control (7.2.1). This conservation law is equivalent to k(IlockΩ + IrotorΩr)− Irotor(Ω + Ωr) = p; that is, (Irotor − kIrotor)Ωr = (kIlock − Irotor)Ω− p. Therefore, if kIlock = Irotor (7.2.9) then one obtains as a special case, the dual spin case in which the acceler- ation feedback is such that each rotor rotates at constant angular velocity relative to the carrier (see Krishnaprasad [1985] and Sa´nchez de Alvarez [1989]). Also note that the Hamiltonian in (7.2.8) can be indefinite. If we set mb = m− ξ, (7.2.4) become m˙b = (mb + ξ)× J−1mb. (7.2.10) It is instructive to consider the case where k = diag(k1, k2, k3). Let I = (Ilock − Irotor) = diag(I˜1, I˜2, I˜3) and the matrix J satisfies the symmetry hypothesis of Theorem 7.2.1. Then li = pi + kimi, i = 1, 2, 3, and the equations become m˙ = m×∇H where H = 1 2 [ ((1− k1)m1 + p1)2 (1− k1)I˜1 + ((1− k2)m2 + p2)2 (1− k2)I˜2 + ((1− k3)m3 + p3)2 (1− k3)I˜3 ] . (7.2.11) 138 7. Stabilization and Control It is possible to have more complex feedback mechanisms where the system still reduces to a Hamiltonian system on so(3)∗. We refer to Bloch, Krish- naprasad, Marsden, and Sanchez de Alvarez [1992] for a discussion of this point. 7.3 Feedback Stabilization of a Rigid Body with a Single Rotor We now consider the equations for a rigid body with a single rotor. We will demonstrate that with a single rotor about the third principal axis, a suitable quadratic feedback stabilizes the system about its intermediate axis. Let the rigid body have moments of inertia I1 > I2 > I3 and suppose the symmetric rotor is aligned with the third principal axis and has moments of inertia J1 = J2 and J3. Let Ωi, i = 1, 2, 3, denote the carrier body angular velocities and let α˙ denote that of the rotor (relative to a frame fixed on the carrier body). Let diag(λ1, λ2, λ3) = diag(J1 + I1, J2 + I2, J3 + I3) (7.3.1) be the (body) locked inertia tensor. Then from (7.1.2) and (7.1.3), the natural momenta are mi = (Ji + Ii)Ωi = λiΩi, i = 1, 2, m3 = λ3Ω3 + J3α˙, (7.3.2) l3 = J3(Ω3 + α˙). Note that m3 = I3Ω3 + l3. The equations of motion (7.1.4) are: m˙1 = m2m3 ( 1 I3 − 1 λ2 ) − l3m2 I3 , m˙2 = m1m3 ( 1 λ1 − 1 I3 ) + l3m1 I3 , m˙3 = m1m2 ( 1 λ2 − 1 λ1 ) , l˙3 = u. (7.3.3) Choosing u = ka3m1m2 where a3 = ( 1 λ2 − 1 λ1 ) , and noting that l3 − km3 = p is a constant, we get: 7.3 Feedback Stabilization of a Rigid Body with a Single Rotor 139 7.3.1 Theorem. With this choice of u and p, the equations (7.3.3) reduce to m˙1 = m2 ( (1− k)m3 − p I3 ) − m3m2 λ2 , m˙2 = −m1 ( (1− k)m3 − p I3 ) + m3m1 λ1 , m˙3 = a3m1m2 (7.3.4) which are Hamiltonian on so(3)∗ with respect to the standard rigid body Lie–Poisson bracket, with Hamiltonian H = 1 2 ( m21 λ1 + m22 λ2 + ((1− k)m3 − p)2 (1− k)I3 ) + 1 2 p2 J3(1− k) (7.3.5) where p is a constant. When k = 0, we get the equations for the rigid body carrying a free spinning rotor—note that this case is not trivial! The rotor interacts in a nontrivial way with the dynamics of the carrier body. We get the dual spin case for which J3α¨ = 0, when kIlock = Irotor from (7.2.9), or, in this case, when k = J3/λ3. Notice that for this k, p = (1− k)α˙, a multiple of α˙. We can now use the energy–Casimir method to prove 7.3.2 Theorem. For p = 0 and k > 1 − (J3/λ2), the system (7.3.4) is stabilized about the middle axis, that is, about the relative equilibrium (0,M, 0). Proof. Consider the energy–Casimir function H+C where C = ϕ(‖m‖2), and m2 = m21 +m 2 2 +m 2 3. The first variation is δ(H + C) = m1δm1 λ1 + m2δm2 λ2 + (1− k)m3 − p I3 δm3 + ϕ′(m2)(m1δm1 +m2δm2 +m3δm3). (7.3.6) This is zero if we choose ϕ so that m1 λ1 + ϕ′m1 = 0, m2 λ2 + ϕ′m2 = 0, (7.3.7) (1− k)m3 − p I3 + ϕ′m3 = 0. Then we compute δ2(H + C) = (δm1)2 λ1 + (δm2)2 λ2 + (1− k)(δm3)2 I3 + ϕ′(m2)((δm1)2 + (δm2)2 + (δm3)2) + ϕ′′(m2)(m1δm1 +m2δm2 +m3δm3)2. (7.3.8) 140 7. Stabilization and Control For p = 0, that is, l3 = km3, (0,M, 0) is a relative equilibrium and (7.3.7) are satisfied if ϕ′ = −1/λ2 at equilibrium. In that case, δ2(H + C) = (δm1)2 ( 1 λ1 − 1 λ2 ) + (δm3)2 ( 1− k I3 − 1 λ2 ) + ϕ′′(δm2)2. Now 1 λ1 − 1 λ2 = I2 − I1 λ1λ2 < 0 for I1 > I2 > I3. For k satisfying the condition in the theorem, 1− k I3 − 1 λ2 < 0, so if one chooses ϕ such that ϕ′′ < 0 at equilibrium, then the second variation is negative definite and hence stability holds. For a geometric interpretation of the stabilization found here, see Holm and Marsden [1991], along with other interesting facts about rigid rotors and pendula, for example, a proof that the rigid body phase space is a union of simple pendulum phase spaces. Corresponding to the Hamiltonian (7.3.5) there is a Lagrangian found using the inverse Legendre transformation Ω˜1 = m1 λ1 , Ω˜2 = m2 λ2 , Ω˜3 = (1− k)m3 − p I3 , ˙˜α = − (1− k)m3 − p (1− k)I3 + p (1− k)I3 . (7.3.9) Note that Ω˜1, Ω˜2 and Ω˜3 equal the angular velocities Ω1,Ω2, and Ω3 for the free system, but that ˙˜α is not equal to α˙. In fact, we have the interesting velocity shift ˙˜α = α˙ (1− k) − km3 (1− k)J3 . (7.3.10) Thus the equations on T SO(3) determined by (7.3.9) are the Euler– Lagrange equations for a Lagrangian quadratic in the velocities, so the equations can be regarded as geodesic equations. The torques can be thought of as residing in the velocity shift (7.3.10). Using the free Lagrangian, the torques appear as generalized forces on the right hand side of the Euler– Lagrange equations. Thus, the d’Alembert principle can be used to describe the Euler–Lagrange equations with the generalized forces. This approach arranges in a different way, the useful fact that the equations are derivable 7.4 Phase Shifts 141 from a Lagrangian (and hence a Hamiltonian) in velocity shifted variables. In fact, it seems that the right context for this is the Routhian and the theory of gyroscopic Lagrangians, as in §3.6, but we will not pursue this further here. For problems like the driven rotor and specifically the dual spin case where the rotors are driven with constant angular velocity, one might think that this is a velocity constraint and should be treated by using constraint theory. For the particular problem at hand, this can be circumvented and in fact, standard methods are applicable and constraint theory is not needed, as we have shown. Finally, we point out that a far reaching generalization of the ideas in the preceding sections for stabilization of mechanical systems has been carried out in the series of papers of Bloch, Leonard and Marsden cited in the references. They use the technique of controlled Lagrangians whereby the system with controls is identified with the Euler-Lagrange equations for a different Lagrangian, namely the controlled Lagrangian. 7.4 Phase Shifts Next, we discuss an attitude drift that occurs in the system and suggest a method for correcting it. If the system (7.2.10) is perturbed from a stable equilibrium, and the perturbation is not too large, the closed loop system executes a periodic motion on a level surface (momentum sphere) of the Casimir function ‖mb+ξ‖2 in the body-rotor feedback system. This leads to an attitude drift which can be thought of as rotation about the (constant) spatial angular momentum vector. We calculate the amount of this rotation following the method of Montgomery we used in Chapter 6 to calculate the phase shift for the single rigid body. As we have seen, the equations of motion for the rigid body-rotor system with feedback law (7.2.1) are m˙b = (mb + ξ)× J−1mb = ∇C ×∇H (7.4.1) where mb = m+ (1− k)−1(km− l), ξ = −(1− k)−1(km− l), and C = 1 2 ‖mb + ξ‖2, (7.4.2) H = 1 2 mb · J−1mb. (7.4.3) As with the rigid body, this system is completely integrable with trajecto- ries given by intersecting the level sets of C and H. Note that (7.4.2) just defines a sphere shifted by the amount ξ. The attitude equation for the rigid-body rotor system is A˙ = AΩˆ, 142 7. Stabilization and Control where ˆ denotes the isomorphism between R3 and so(3) (see §4.4) and in the presence of the feedback law (7.2.1), Ω = (Ilock − Irotor)−1(m− l) = (Ilock − Irotor)−1(m− km+ p) = J−1(m− ξ) = J−1mb. Therefore the attitude equation may be written A˙ = A(J−1mb)ˆ. (7.4.4) The net spatial (constant) angular momentum vector is µ = A(mb + ξ). (7.4.5) Then we have the following: 7.4.1 Theorem. Suppose the solution of m˙b = (mb + ξ)× J−1mb (7.4.6) is a periodic orbit of period T on the momentum sphere, ‖mb + ξ‖2 = ‖µ‖2, enclosing a solid angle Φsolid. Let Ωav denote the average value of the body angular velocity over this period, E denote the constant value of the Hamiltonian, and ‖µ‖ denote the magnitude of the angular momentum vector. Then the body undergoes a net rotation ∆θ about the spatial angular momentum vector µ given by ∆θ = 2ET ‖µ‖ + T ‖µ‖ (ξ · Ωav)− Φsolid. (7.4.7) Proof. Consider the reduced phase space (the momentum sphere) Pµ = {mb | ‖mb + ξ‖2 = ‖µ‖2 }, (7.4.8) for µ fixed, so that in this space, mb(t0 + T ) = mb(t0), (7.4.9) and from momentum conservation µ = A(T + t0)(mb(T + t0) + ξ) = A(t0)(mb(t0) + ξ). (7.4.10) Hence A(T + t0)A(t0)−1µ = µ. Thus A(T + t0)A(t0)−1 ∈ Gµ, so A(T + t0)A(t0)−1 = exp ( ∆θ µ ‖µ‖ ) (7.4.11) 7.4 Phase Shifts 143 for some ∆θ, which we wish to determine. We can assume that A(t0) is the identity, so the body is in the reference configuration at t0 = 0 and thus that mb(t0) = µ − ξ. Consider the phase space trajectory of our system z(t) = (A(t),mb(t)), z(t0) = z0. (7.4.12) The two curves in phase space C1 = { z(t) | t0 ≤ t ≤ t0 + T } (the dynamical evolution from z0), and C2 = { exp ( θ µ ‖µ‖ ) z0 ∣∣∣∣ 0 ≤ θ ≤ ∆θ} intersect at t = T . Thus C = C1 −C2 is a closed curve in phase space and so by Stokes’ theorem,∫ C1 pdq − ∫ C2 pdq = ∫∫ Σ d(pdq) (7.4.13) where pdq = ∑3 i=1 pidqi and where qi and pi are configuration space vari- ables and conjugate momenta in the phase space and Σ is a surface enclosed by the curve C.1 Evaluating each of these integrals will give the formula for ∆θ. Letting ω be the spatial angular velocity, we get p dq dt = µ · ω = A(JΩ + ξ) ·AΩ = JΩ · Ω + ξ · Ω. (7.4.14) Hence∫ C1 pdq = ∫ C1 p · dq dt dt = ∫ T 0 JΩ · Ωdt+ ∫ T 0 ξ · Ωdt = 2ET + (ξ · Ωav)T (7.4.15) since the Hamiltonian is conserved along orbits. Along C2,∫ C2 p · dq dt dt = ∫ C2 µ · ωdt = ∫ C2 µ · { dθ dt µ ‖µ‖ } dt = ‖µ‖ ∫ C2 dθ = ‖µ‖∆θ. (7.4.16) Finally we note that the map piµ from the set of points in phase space with angular momentum µ to Pµ satisfies∫∫ Σ d(pdq) = ∫∫ piµ(Σ) dA = ‖µ‖Φsolid (7.4.17) 1Either one has to show such a surface exists (as Montgomery has done) or, alter- natively, one can use general facts about holonomy (as in Marsden, Montgomery, and Ratiu [1990]) that require only a bounding surface in the base space, which is obvious in this case. 144 7. Stabilization and Control where dA is the area form on the two-sphere and piµ(Σ) is the spherical cap bounded by the periodic orbit {mb(t) | t0 ≤ t ≤ t0 + T } ⊂ Pµ. Combining (7.4.15)–(7.4.17) we get the result. Remarks. 1. When ξ = 0, (7.4.7) reduces to the Goodman–Robinson–Montgomery formula in Chapter 6. 2. This theorem may be viewed as a special case of a scenario that is use- ful for other systems, such as rigid bodies with flexible appendages. As we saw in Chapter 6, phases may be viewed as occurring in the reconstruction process, which lifts the dynamics from Pµ to J−1(µ). By the cotangent bundle reduction theorem, Pµ is a bundle over T ∗S, where S = Q/G is shape space. The fiber of this bundle is Oµ, the coadjoint orbit through µ. For a rigid body with three internal rotors, S is the three torus T3 parametrized by the rotor angles. Controlling them by a feedback or other control and using other conserved quan- tities associated with the rotors as we have done, leaves one with dynamics on the “rigid variables” Oµ, the momentum sphere in our case. Then the problem reduces to that of lifting the dynamics on Oµ to J−1(µ) with the T ∗S dynamics given. For G = SO(3) this “re- duces” the problem to that for geometric phases for the rigid body. 3. Some interesting control maneuvers for “satellite parking” using these ideas may be found in Walsh and Sastry [1993]. 4. See Montgomery [1990, p. 569] for comments on the Chow–Ambrose– Singer theorem in this context. Finally, following a suggestion of Krishnaprasad, we show that in the zero total angular momentum case one can compensate for this drift using two rotors. The total spatial angular momentum if one has only two rotors is of the form µ = A(IlockΩ + b1α˙1 + b2α˙2) (7.4.18) where the scalars α˙1 and α˙2 represent the rotor velocities relative to the body frame. The attitude matrix A satisfies A˙ = AΩˆ, (7.4.19) as above. If µ = 0, then from (7.4.18) and (7.4.19) we get, A˙ = −A((I−1lockb1)ˆ α˙1 + (I−1lockb2)ˆ α˙2). (7.4.20) 7.5 The Kaluza–Klein Description of Charged Particles 145 It is well known (see for instance Brockett [1973] or Crouch [1986]) that if we treat the α˙i, i = 1, 2 as controls, then attitude controllability holds iff (I−1lockb1)ˆ and (I −1 lockb2)ˆ generate so(3), or equivalently, iff the vectors I−1lockb1 and I −1 lockb2 are linearly independent. (7.4.21) Moreover, one can write the attitude matrix as a reverse path-ordered exponential A(t) = A(0)·P¯ exp [ − ∫ t 0 { (I−1lockb1)ˆ α˙1(σ) + (I −1 lockb2)ˆ α˙2(σ) } dσ ] . (7.4.22) The right hand side of (7.4.22) depends only on the path traversed in the space T2 of rotor angles (α1, α2) and not on the history of velocities α˙i. Hence the Formula (7.4.22) should be interpreted as a “geometric phase”. Furthermore, the controllability condition can be interpreted as a curvature condition on the principal connection on the bundle T2 × SO(3) → T2 defined by the so(3)-valued differential 1-form, θ(α1, α2) = −((I−1lockb1)ˆdα1 + (I−1lockb2)ˆdα2). (7.4.23) For further details on this geometric picture of multibody interaction see Krishnaprasad [1989] and Wang and Krishnaprasad [1992]. In fact, our discussion of the two coupled bodies in §6.1 can be viewed as an especially simple planar version of what is required. 7.5 The Kaluza–Klein Description of Charged Particles In preparation for the next section we describe the equations of a charged particle in a magnetic field in terms of geodesics. The description we saw in §2.10 can be obtained from the Kaluza–Klein description using an S1- reduction. The process described here generalizes to the case of a particle in a Yang–Mills field by replacing the magnetic potential A by the Yang–Mills connection. We are motivated as follows: since charge is a conserved quantity, we introduce a new cyclic variable whose conjugate momentum is the charge. This process is applicable to other situations as well; for example, in fluid dynamics one can profitably introduce a variable conjugate to the conserved mass density or entropy; cf. Marsden, Ratiu, and Weinstein [1984a, 1984b]. For a charged particle, the resultant system is in fact geodesic motion! 146 7. Stabilization and Control Recall from §2.10 that if B = ∇ × A is a given magnetic field on R3, then with respect to canonical variables (q,p), the Hamiltonian is H(q,p) = 1 2m ‖p− e c A‖2. (7.5.1) We can obtain (7.5.1) via the Legendre transform if we choose L(q, q˙) = 1 2 m‖q˙‖2 + e c A · q˙ (7.5.2) for then p = ∂L ∂q˙ = mq˙ + e c A (7.5.3) and p · q˙− L(q, q˙) = ( mq˙ + e c A ) · q˙− 1 2 m‖q˙‖2 − e c A · q˙ = 1 2 m‖q˙‖2 = 1 2m ‖p− e c A‖2 = H(q,p). (7.5.4) Thus, the Euler–Lagrange equations for (7.5.2) reproduce the equations for a particle in a magnetic field. (If an electric field E = −∇ϕ is present as well, subtract eϕ from L, treating eϕ as a potential energy.) Let the Kaluza–Klein configuration space be QK = R3 × S1 (7.5.5) with variables (q, θ) and consider the one-form ω = A+ dθ (7.5.6) on QK regarded as a connection one-form. Define the Kaluza–Klein La- grangian by LK(q, q˙, θ, θ˙) = 1 2 m‖q˙‖2 + 1 2 ‖〈ω, (q, q˙, θ, θ˙)〉‖2 = 1 2 m‖q˙‖2 + 1 2 (A · q˙ + θ˙)2. (7.5.7) The corresponding momenta are p = mq˙ + (A · q˙ + θ˙)A and pθ = A · q˙ + θ˙. (7.5.8) Since (7.5.7) is quadratic and positive definite in q˙ and θ˙, the Euler– Lagrange equations are the geodesic equations on R3×S1 for the metric for which LK is the kinetic energy . Since pθ is constant in time as can be seen 7.5 The Kaluza–Klein Description of Charged Particles 147 from the Euler–Lagrange equation for (θ, θ˙), we can define the charge e by setting pθ = e/c; (7.5.9) then (7.5.8) coincides with (7.5.3). The corresponding Hamiltonian on T ∗QK endowed with the canonical symplectic form is HK(q, p, θ, pθ) = 1 2 m‖p− pθA‖2 + 12p 2 θ. (7.5.10) Since pθ is constant, HK differs from H only by the constant p2θ/2. Notice that the reduction of the Kaluza–Klein system by S1 reproduces the description in terms of (7.5.1) or (7.5.2). The magnetic terms in the sense of reduction become the magnetic terms we started with. Note that the description in terms of the Routhian from §3.6 can also be used here, reproducing the same results. Also notice the similar way that the gyro- scopic terms enter into the rigid body system with rotors—they too can be viewed as magnetic terms obtained through reduction. A Particle in a Yang–Mills Field. These Kaluza-Klein constructions generalize to the case of a particle in a Yang–Mills field where ω becomes the connection of a Yang–Mills field and its curvature measures the field strength which, for an electromagnetic field, reproduces the relation B = ∇×A. Some of the key literature on this topic is Kerner [1968], Wong [1970], Sternberg [1977], Weinstein [1978a] and the beautiful work of Montgomery [1984, 1986]. The equations for a particle in a Yang-Mills field, often called Wong’s equations, involve a given principal connection A (a Yang-Mills field) on a principal bundle pi : Q → S = Q/G. As is well known, one can write a principal connection in a local trivialization Q = S × G, with coordinates (xα, ga) as A(x, g)(x˙, g˙) = Adg(g−1g˙ +Aloc(x)x˙). and the components of the connection are defined by Aloc(x) · v = Aaαvαea where ea is a basis of g, and the curvature is the g-valued two-form given by the standard formula Bbαβ = ∂Abα ∂rβ − ∂A b β ∂rα − CbacAaβAcα. Wong’s equations in coordinates are as follows: p˙α = −λaBaαβ x˙β − 1 2 ∂gβγ ∂xα pβpγ (7.5.11) λ˙b = −λaCadbAdαx˙α (7.5.12) 148 7. Stabilization and Control where gαβ is the local representation of the metric on the base space S, gβγ is the inverse of the matrix gαβ , and pα is defined by pα = gαβ x˙β . These equations have an intrinsic description as Lagrange–Poincare´ equa- tions on the reduced tangent bundle given by (TQ)/G ∼= T (Q/G) × g˜ or its Hamiltonian counterpart, (T ∗Q)/G ∼= T ∗(Q/G)× g˜∗. We leave it to the reader to consult these references, especially those of Montgomery as well as Cendra, Marsden, and Ratiu [2001], Chapter 4, for an intrinsic account of these equations and their relation to reduction. Finally, we remark that the relativistic context is the most natural in which to introduce the full electromagnetic field. In that setting the con- struction we have given for the magnetic field will include both electric and magnetic effects. Consult Misner, Thorne, and Wheeler [1973], Gotay and Marsden [1992] and Marsden and Shkoller [1998] for additional information. 7.6 Optimal Control and Yang–Mills Particles This section discusses an elegant link between optimal control and the dynamics of a particle in a Yang–Mills field that was discovered by Mont- gomery, who was building on work of Shapere and Wilczeck [1989]. We refer to Montgomery [1990] for further details and references. This topic, together with the use of connections described in previous chapters has lead one to speak about the “gauge theory of deformable bodies”. The example of a falling cat as a control system is good to keep in mind while reading this section. We start with a configuration space Q and assume we have a symmetry group G acting freely by isometries, as before. Put on the bundle Q→ S = Q/G the mechanical connection, as described in §3.3. Fix a point q0 ∈ Q and a group element g ∈ G. Let hor(q0, gq0) = all horizontal paths joining q0 to gq0. (7.6.1) This space of (suitably differentiable) paths may be regarded as the space of horizontal paths with a given holonomy g. Recall from §3.3 that horizontal means that this path is one along which the total angular momentum (i.e., the momentum map of the curve in T ∗Q corresponding to q˙) is zero. The projection of the curves in hor (q0, gq0) to shape space S = Q/G are closed, as in Figure 7.6. The optimal control problem we wish to discuss is: Find a path c in hor(q0, gq0) whose base curve x has minimal (or extremal) kinetic energy . The minimum (or critical point) is taken over all paths s obtained from projections of paths in hor(q0, gq0). 7.6 Optimal Control and Yang–Mills Particles 149 Q S G -orbit Figure 7.6.1. Loops in S with a fixed holonomy g and base point q0. One may wish to use functions other than the kinetic energy of the shape space path s to extremize. For example, the cat may wish to turn itself over by minimizing some combination of the time taken and the amount of work it actually does. 7.6.1 Theorem. Consider a path c ∈ hor(q0, gq0) with given holonomy g and (with shape space curve denoted by x ∈ S). If c is extremal, then there is a curve λ in g∗ such that the pair (x, λ) satisfies Wong’s equations, 7.5.11 and 7.5.12. This result is closely related to, and may be deduced from, our work on Lagrangian reduction in §3.6. The point is that c ∈ hor(q0, gq0) means c˙ ∈ J−1(0), and what we want to do is relate a variational problem on Q but with the constraint J−1(0) imposed and with Hamiltonian the Kaluza– Klein kinetic energy on TQ with a reduced variational principle on S. This is exactly the set up to which the discussion in §3.6 applies. More precisely, from considerations in the Calculus of Variations, if one has a constrained critical point, there is a curve λ(t) ∈ g∗ such that the unconstrained cost function C(q, q˙) = ∫ ( 1 2 ‖Hor(q, q˙)‖2 + 〈λ,Amec(q, q˙)〉 ) dt has a critical point at q(t) (use 〈ξ,J〉 if one wishes to replace λ with a curve ξ in the Lie algebra—λ and ξ will be related by the locked inertia tensor). This cost function is G-invariant and the methods of Lagrangian reduction 150 7. Stabilization and Control can be applied to it. One finds that the corresponding Lagrange–Poincare´ equations (that is, the reduced Euler–Lagrange equations) are exactly the Wong equations. For details of this compuation, see Koon and Marsden [1997a], Cendra, Holm, Marsden, and Ratiu [1998] or Bloch [2003]. What is less obvious is how to use the methods of the calculus of vari- ations to show the existence of minimizing loops with a given holonomy and what their smoothness properties are. For this purpose, the subject of sub-Riemannian geometry (dealing with degenerate metrics) is relevant. The shape space kinetic energy, thought of as ‖Hor(vq)‖2 and regarded as a quadratic form on TQ (rather that TS), is associated with a degenerate metric. These are sometimes called Carnot–Caratheodory metrics. We refer to Montgomery [1990, 1991, 2002]] for further discussion. Montgomery’s theorem has been generalized to the case of nonzero mo- mentum values as well as to nonholonomic systems in Koon and Marsden [1997a] by making use of the techniques of Lagrangian reduction. This original proof, as well as related problems (such as the rigid body with two oscillators considered by Yang, Krishnaprasad, and Dayawansa [1996], and variants of the falling cat problem in Enos [1993]) were done by means of the Pontryagin principle together with Poisson reduction. The process of Lagrangian reduction appears, at least to some, to be somewhat simpler and more direct—it was this method that produced the generalizations of the result mentioned just above. Page 151 8 Discrete Reduction In this chapter, we extend the theory of reduction of Hamiltonian systems with symmetry to include systems with a discrete symmetry group acting symplectically .1 For antisymplectic symmetries such as reversibility, this question has been considered by Meyer [1981] and Wan [1990]. However, in this chapter we are concerned with symplectic symmetries. Antisymplectic symmetries are typified by time reversal symmetry, while symplectic symmetries are typified by spatial discrete symmetries of systems like reflection symmetry. Often these are obtained by taking the cotangent lift of a discrete symmetry of configuration space. There are two main motivations for the study of discrete symmetries. The first is the theory of bifurcation of relative equilibria in mechanical systems with symmetry. The rotating liquid drop is a system with a sym- metric relative equilibrium that bifurcates via a discrete symmetry. An initially circular drop (with symmetry group S1) that is rotating rigidly in the plane with constant angular velocity Ω, radius r, and with surface tension τ , is stable if r3Ω2 < 12τ (this is proved by the energy–Casimir or energy–momentum method). Another relative equilibrium (a rigidly ro- tating solution in this example) branches from this circular solution at the critical point r3Ω2 = 12τ . The new solution has the spatial symmetry of an ellipse; that is, it has the symmetry Z2 × Z2 (or equivalently, the dihe- 1Parts of the exposition in this lecture are based on unpublished work with J. Harnad and J. Hurtubise. 152 8. Discrete Reduction dral group D2). These new solutions are stable, although whether they are subcritical or supercritical depends on the parametrization chosen (angular velocity vs. angular momentum, for example). This example is taken from Lewis, Marsden, and Ratiu [1987] and Lewis [1989]. It also motivated some of the work in the general theory of bifurcation of equilibria for Hamiltonian systems with symmetry by Golubitsky and Stewart [1987]. The spherical pendulum is an especially simple mechanical system having both continuous (S1) and discrete (Z2) symmetries. Solutions can be rela- tive equilibria for the action of rotations about the axis of gravity in which the pendulum rotates in a circular motion. For zero angular momentum these solutions degenerate to give planar oscillations, which lie in the fixed point set for the action of reflection in that plane. In our approach, this reflection symmetry is cotangent lifted to produce a symplectic involution of phase space. The invariant subsystem defined by the fixed point set of this involution is just the planar pendulum. The double spherical pendulum has the same continuous and discrete symmetry group as the spherical pendulum. However, the double spherical pendulum has another nontrivial symmetry involving a spacetime symme- try, much as the rotating liquid drop and the water molecule. Namely, we are at a fixed point of a symmetry when the two pendula are rotating in a steadily rotating vertical plane—the symmetry is reflection in this plane. This symmetry is, in fact, the source of the subblocking property of the second variation of the amended potential that we observed in Chapter 5. In the S1 reduced space, a steadily rotating plane is a stationary plane and the discrete symmetry just becomes reflection in that plane. A related example is the double whirling mass system. This consists of two masses connected to each other and to two fixed supports by springs, but with no gravity. Assume the masses and springs are identical. Then there are two Z2 symmetries now, corresponding to reflection in a (steadily rotating) vertical plane as with the double spherical pendulum, and to swapping the two masses in a horizontal plane. The classical water molecule has a discrete symplectic symmetry group Z2 as well as the continuous symmetry group SO(3). The discrete symmetry is closely related to the symmetry of exchanging the two hydrogen atoms. For these examples, there are some basic links to be made with the block diagonalization work from Chapter 5. In particular, we show how the dis- crete symmetry can be used to refine the block structure of the second variation of the augmented Hamiltonian and of the symplectic form. Recall that the block diagonalization method provides coordinates in which the second variation of the amended potential on the reduced configuration space is a block diagonal matrix, with the group variables separated from the internal variables; the group part corresponds to a bilinear form com- puted first by Arnold for purposes of examples whose configuration space is a group. The internal part corresponds to the shape space variables, that are on Q/G the quotient of configuration space by the continuous sym- 8.1 Fixed Point Sets and Discrete Reduction 153 metry group. For the water molecule, we find that the discrete symmetry provides a further blocking of this second variation, by splitting the internal tangent space naturally into symmetric modes and nonsymmetric ones. In the dynamics of coupled rigid bodies one has interesting symmetry breaking bifurcations of relative equilibria and of relative periodic orbits. For the latter, discrete spacetime symmetries are important. We refer to Montaldi, Roberts, and Stewart [1988], Oh, Sreenath, Krishnaprasad, and Marsden [1989] and to Patrick [1989, 1990] for further details. In the opti- mal control problem of the falling cat (Montgomery [1990] and references therein), the problem is modeled as the dynamics of two identical coupled rigid bodies and the fixed point set of the involution that swaps the bodies describes the “no-twist” condition of Kane and Scher [1969]; this plays an essential role in the problem and the dynamics is integrable on this set. The second class of examples motivating the study of discrete symmetries are integrable systems, including those of Bobenko, Reyman, and Semenov- Tian-Shansky [1989]. This (spectacular) reference shows in particular how reduction and dual pairs, together with the theory of R-matrices, can be used to understand the integrability of a rich class of systems, including the celebrated Kowalewski top. They also obtain all of the attendant al- gebraic geometry in this context. It is clear from their work that discrete symmetries, and specifically those obtained from Cartan involutions, play a crucial role. 8.1 Fixed Point Sets and Discrete Reduction Let (P,Ω) be a symplectic manifold, G a Lie group acting on P by symplec- tic transformations and J : P → g∗ an Ad∗-equivariant momentum map for the G-action. Let Σ be a compact Lie group acting on P by symplectic transformations and by group homomorphisms on G. For σ ∈ Σ, write σP : P → P and σG : G→ G for the corresponding symplectic map on P and group homomorphism on G. Let σg : g→ g be the induced Lie algebra homomorphism (the derivative of σG at the identity) and σg∗ : g∗ → g∗ be the dual of (σ−1)g, so σg∗ is a Poisson map with respect to the Lie–Poisson structure. Remarks. 1. One can also consider the Poisson case directly using the methods of Poisson reduction (Marsden and Ratiu [1986]) or by considering the present work applied to their symplectic leaves. 154 8. Discrete Reduction 2. As in the general theory of equivariant momentum maps, one may drop the equivariance assumption by using another action on g∗. 3. Compactness of Σ is used to give invariant metrics obtained by aver- aging over Σ—it can be weakened to the existence of invariant finite measures on Σ. 4. The G-action will be assumed to be a left action, although right actions can be treated in the same way. Assumption 1. The actions of G and of Σ are compatible in the sense that the following equation holds: σP ◦ gP = [σG(g)]P ◦ σP (8.1.1) for each σ ∈ Σ and g ∈ G, where we have written gP for the action of g ∈ G on P . (See Figure 8.1.1.) P P P P gP [σG(g)]P σP σP - - ? ? Figure 8.1.1. Assumption 1. If we differentiate Equation (8.1.1) with respect to g at the identity g = e, in the direction ξ ∈ g, we get TσP ◦X〈J,ξ〉 = X〈J,σg·ξ〉 ◦ σP (8.1.2) where Xf is the Hamiltonian vector field on P generated by the function f : P → R. Since σP is symplectic, (8.1.2) is equivalent to X〈J,ξ〉◦σ−1P = X〈J,σg·ξ〉; (8.1.3) that is, J ◦ σP = σg∗ ◦ J + (cocycle). (8.1.4) We shall assume, in addition, that the cocycle is zero. In other words, we make: 8.1 Fixed Point Sets and Discrete Reduction 155 P g∗ P g∗ J J σP σg∗ - - ? ? Figure 8.1.2. Assumption 2. Assumption 2. The following equation holds (see Figure 8.1.2): J ◦ σP = σg∗ ◦ J. (8.1.5) Let GΣ = Fix(Σ, G) ⊂ G be the fixed point set of Σ; that is, GΣ = { g ∈ G | σG(g) = g for all σ ∈ Σ }. (8.1.6) The Lie algebra of GΣ is the fixed point (i.e., eigenvalue 1) subspace: gΣ = Fix(Σ, g) = { ξ ∈ g | TσG(e) · ξ = ξ for all σ ∈ Σ }. (8.1.7) Remarks. 1. If G is connected, then (8.1.5) (or (8.1.4)) implies (8.1.1). 2. If Σ is a discrete group, then Assumptions 1 and 2 say that J : P → g∗ is an equivariant momentum map for the semi-direct product ΣsG, which has the multiplication (σ1, g1) · (σ2, g2) = (σ1σ2, (g2 · (σ2)G(g1))). (8.1.8) Recall that Σ embeds as a subgroup of ΣsG via σ 7→ (σ, e). The action of Σ on G is given by conjugation within ΣsG. If one prefers, one can take the point of view that one starts with a group G with G chosen to be the connected component of the identity and with Σ a subgroup of G isomorphic to the quotient of G by the normal subgroup G. 3. One checks that ΣsGΣ = N(Σ), where N(Σ) is the normalizer of Σ × {e} within G = ΣsG. Accordingly, we can identify GΣ with the quotient N(Σ)/Σ. In Golubitsky and Stewart [1987] and related works, the tendency is to work with the group N(Σ)/Σ; it will be a bit more convenient for us to work with the group GΣ, although the two approaches are equivalent, as we have noted. 156 8. Discrete Reduction Let PΣ = Fix(Σ, P ) ⊂ P be the fixed point set for the action of Σ on P : PΣ = { z ∈ P | σP (z) = z for all σ ∈ Σ }. (8.1.9) 8.1.1 Proposition. The fixed point set PΣ is a smooth symplectic sub- manifold of P . Proof. Put on P a metric that is Σ-invariant and exponentiate the linear fixed point set (TzP )Σ of the tangent map Tzσ : TzP → TzP for each z ∈ PΣ to give a local chart for PΣ. Since (TzP )Σ is invariant under the associated complex structure, it is symplectic, so PΣ is symplectic. 8.1.2 Proposition. The manifold PΣ is invariant under the action of GΣ. Proof. Let z ∈ PΣ and g ∈ GΣ. To show that gP (z) ∈ PΣ, we show that σP (gP (z)) = gP (z) for σ ∈ Σ. To see this, note that by (8.1.1), and the facts that σG(g) = g and σP (z) = z, σP (gP (z)) = [σG(g)]P (σP (z)) = gP (z). The main message of this section is the following: Discrete Reduction Procedure. If H is a Hamiltonian system on P that is G and Σ invariant, then its Hamiltonian vector field XH (or its flow) will leave PΣ invariant and so standard symplectic reduction can be performed with respect to the action of the symmetry group GΣ on PΣ. We are not necessarily advocating that one should always perform this discrete reduction procedure, or that it contains in some sense equivalent information to the original system, as is the case with continuous reduction. We do claim that the procedure is an interesting way to identify invariant subsystems, and to generate new ones that are important in their own right. For example, reducing the spherical pendulum by discrete reflection in a plane gives the half-dimensional simple planar pendulum. Discrete reduc- tion of the double spherical pendulum by discrete symmetry produces a planar compound pendulum, possibly with magnetic terms. More impres- sively, the Kowalewski top arises by discrete reduction of a larger system. Also, the ideas of discrete reduction are very useful in the block diagonal- ization stability analysis of the energy momentum method, even though the reduction is not carried out explicitly. To help understand the reduction of PΣ, we consider some things that are relevant for the momentum map of the action of GΣ. The following prepatory lemma is standard (see, for example, Guillemin and Prato [1990]) but we shall need the proof for later developments, so we give it. 8.1 Fixed Point Sets and Discrete Reduction 157 8.1.3 Lemma. The Lie algebra g of G splits as follows: g = gΣ ⊕ (g∗Σ)0 (8.1.10) where g∗Σ = Fix(Σ, g ∗) is the fixed point set for the action of Σ on g∗ and where the superscript zero denotes the annihilator in g. Similarly, the dual splits as g∗ = g∗Σ ⊕ (gΣ)0. (8.1.11) Proof. First, suppose that ξ ∈ gΣ ∩ (g∗Σ)0, µ ∈ g∗ and σ ∈ Σ. Since ξ ∈ gΣ, 〈µ, ξ〉 = 〈µ, σg · ξ〉 = 〈σ−1g∗ · µ, ξ〉. (8.1.12) Averaging (8.1.12) over σ relative to an invariant measure on Σ, which is possible since Σ is compact, gives: 〈µ, ξ〉 = 〈µ¯, ξ〉. Since the average µ¯ of µ is Σ-invariant, we have µ¯ ∈ g∗Σ and since ξ ∈ (g∗Σ)0 we get 〈µ, ξ〉 = 〈µ¯, ξ〉 = 0. Since µ was arbitrary, ξ = 0. Thus, gΣ ∩ (g∗Σ)0 = {0}. Using an invariant metric on g, we see that dim gΣ = dim(g∗Σ) 0 and thus dim g = dim gΣ + dim(g∗Σ) 0, so we get the result (8.1.10). The proof of (8.1.11) is similar. (If the group is infinite dimensional, then one needs to show that every element can be split; in practice, this usually relies on a Fredholm alternative-elliptic equation argument.) It will be useful later to note that this argument also proves a more general result as follows: 8.1.4 Lemma. Let Σ act linearly on a vector space W . Then W = WΣ ⊕ (W ∗Σ)0 (8.1.13) where WΣ is the fixed point set for the action of Σ on W and where (W ∗Σ) 0 is the annihilator of the fixed point set for the dual action of Σ on W ∗. Similarly, the dual splits as W ∗ = W ∗Σ ⊕ (WΣ)0. (8.1.14) Adding these two splittings gives the splitting W ×W ∗ = (WΣ ×W ∗Σ)⊕ ((W ∗Σ)0 × (WΣ)0) (8.1.15) which is the splitting of the symplectic vector space W × W ∗ into the symplectic subspace WΣ ×W ∗Σ and its symplectic orthogonal complement. The splitting (8.1.10) gives a natural identification (gΣ)∗ ∼= g∗Σ. Note that the splittings (8.1.10) and (8.1.11) do not involve the choice of a metric; that is, the splittings are natural. The following block diagonalization into isotypic components will be used to prove the subblocking theorem in the energy–momentum method. 158 8. Discrete Reduction 8.1.5 Lemma. Let B : W ×W → R be a bilinear form invariant un- der the action of Σ. Then the matrix of B block diagonalizes under the splitting (8.1.13). That is, B(w, u) = 0 and B(u,w) = 0 if w ∈WΣ and u ∈ (W ∗Σ)0. Proof. Consider the element α ∈ W ∗ defined by α(v) = B(w, v). It suffices to show that α is fixed by the action of Σ on W ∗ since it annihilates W ∗Σ. To see this, note that invariance of B means B(σw, σv) = B(w, v), that is, B(w, σv) = B(σ−1w, v). Then (σ∗α)(v) = α(σv) = B(w, σv) = B(σ−1w, v) = B(w, v) = α(v) since w ∈ WΣ so w is fixed by σ. This shows that α ∈ W ∗Σ and so α(u) = B(w, u) = 0 for u in the annihilator. The proof that B(u,w) = 0 is similar. For the classical water molecule,WΣ corresponds to symmetric variations of the molecule’s configuration and (W ∗Σ) 0 to non-symmetric ones. 8.1.6 Proposition. The inclusion J(PΣ) ⊂ g∗Σ holds and the momentum map for the GΣ action on PΣ equals J restricted to PΣ, and takes values in g∗Σ ∼= (gΣ)∗. Proof. Let σ ∈ Σ and z ∈ PΣ. By (8.1.5), gg∗ · J(z) = J(σP (z)) = J(z), so J(z) ∈ Fix(Σ, g∗). The second result follows since the momentum map for GΣ acting on PΣ is J composed with the projection of g∗ to g∗Σ. Next, we consider an interesting transversality property of the fixed point set. Assume that each µ ∈ g∗Σ is a regular value of J. Then WΣ = J−1(g∗Σ) ⊂ P is a submanifold, PΣ ⊂WΣ, and pi = J|WΣ : WΣ → gΣ is a submersion. 8.1.7 Proposition. The manifold PΣ is transverse to the fibers J−1(µ)∩ WΣ. Proof. Recall that PΣ is a manifold with tangent space at z ∈ PΣ given by TzPΣ = { v ∈ TzP | TzσP · b = v for all σ ∈ Σ}. Choose ν ∈ g∗Σ and u ∈ TzWΣ such that Tpi · u = ν. Note that if σ ∈ Σ, then Tpi ·Tσ ·u = σg∗Tpi ·u = σg∗ ·ν = ν, so Tσ ·u has the same projection as u. Thus, if we average over Σ, we get u¯ ∈ TzWΣ also with Tpi · u¯ = ν. But since u¯ is the average, it is fixed by each Tzσ, so u¯ ∈ TzPΣ. Thus, TzPΣ is transverse to the fibers of pi. 8.2 Cotangent Bundles 159 See Lemma 3.2.3 of Sjamaar [1990] for a related result. In particular, this result implies that 8.1.8 Corollary. The set of ν ∈ g∗Σ for which there is a Σ-fixed point in J−1(ν), is open. 8.2 Cotangent Bundles Let P = T ∗Q and assume that Σ and G act on Q and hence on T ∗Q by cotangent lift. Assume that Σ acts on G but now assume the actions are compatible in the following sense: Assumption 1Q. The following equation holds (Figure 8.2.1): σQ ◦ gQ = [σG(g)]G ◦ σQ. (8.2.1) Q Q Q Q gQ [σG(g)]Q σQ σQ - - ? ? Figure 8.2.1. Assumption 1Q. 8.2.1 Proposition. Under Assumption 1Q, both Assumptions 1 and 2 are valid. Proof. Equation (8.1.1) follows from (8.2.1) and the fact that cotangent lift preserves compositions. Differentiation of (8.2.1) with respect to g at the identity in the direction ξ ∈ g gives TσQ ◦ ξQ = (σg · ξ)Q ◦ σQ. (8.2.2) Evaluating this at q, pairing the result with a covector p ∈ T ∗σQ(q)Q, and using the formula for the momentum map of a cotangent lift gives 〈p, TσQ · ξQ(q)〉 = 〈p, (σg · ξ)Q(σQ(q))〉 that is, 〈Tσ∗Q · p, ξQ(q)〉 = 〈p, (σg · ξ)Q(σQ(q))〉 that is, 〈J(Tσ∗Q · p), ξ〉 = 〈J(p), σg · ξ〉 = 〈σ−1g∗ J(p), ξ〉 that is, J ◦ σ−1P = σ−1g∗ ◦ J, so (8.1.5) holds. 160 8. Discrete Reduction 8.2.2 Proposition. For cotangent lifts, and QΣ = Fix(Σ, Q), we have PΣ = T ∗(QΣ) (8.2.3) with the canonical cotangent structure. Moreover, the action of GΣ is the cotangent lift of its action on QΣ and its momentum map is the standard one for cotangent lifts. For (8.2.3) to make sense, we need to know how T ∗qQΣ is identified with a subspace of T ∗qQ. To see this, consider the action of Σ on T ∗ qQ by σ · αq = ( [TσQ(q)]−1 )∗ · αq and regard Fix(Σ, T ∗qQ) as a linear subspace of T ∗ qQ. 8.2.3 Lemma. TqQ = TqQΣ ⊕ [Fix(Σ, T ∗qQ)]0. (8.2.4) Proof. As before, TqQΣ = Fix(Σ, TqQ). We prove the lemma by a pro- cedure similar to Lemma 8.1.3. Let v ∈ TqQΣ ∩ Fix(Σ, T ∗qQ)0, ξ ∈ T ∗qQ, and σ ∈ Σ. Then 〈α, v〉 = 〈α, TσQ · v〉 = 〈Tσ−1Q · α, v〉. (8.2.5) Averaging (8.2.5) over σ gives 〈σ, v〉 = 〈α¯, v〉 = 0 so v = 0. The result follows by a dimension count, as in Lemma 8.1.3. If αq ∈ T ∗qQΣ, extend it to T ∗qQ by letting it be zero on [Fix(Σ, T ∗qQ)]0. This embeds T ∗qQΣ into T ∗ qQ and provides the split T ∗qQ = T ∗ qQΣ ⊕ (TqQΣ)0 (8.2.6) identifying T ∗qQΣ ∼= Fix(Σ, T ∗qQ). Returning to the proof of Proposition 8.2.2, by definition of PΣ, αq ∈ PΣ iff σP (αq) = αq for all σ ∈ Σ; that is,( [TσQ(q)]−1 )∗ αq = αq that is, αq ∈ Fix(Σ, T ∗qQ) = T ∗QΣ. Thus, PΣ = T ∗(QΣ). The symplectic structure on PΣ is obtained from T ∗Q by restriction. We need to show that the inclusion map iΣ : T ∗QΣ → T ∗Q defined by (8.2.6) is a symplectic embedding. In fact, it is readily checked that iΣ pulls the canonical one form on T ∗Q back to that on T ∗QΣ because the projection piΣ : T ∗Q→ P satisfies pi ◦ iΣ = piΣ. The rest of the proposition now follows. 8.3 Examples 161 8.3 Examples A. Let the phase space be P = C2 with the symplectic structure given by Ω((z1, w1), (z2, w2)) = 2 Im(z1z¯2 + w1w¯2). Let G = U(2) act simply by matrix multiplication and let Σ = Z2 act on P by σ(z, w) = (w, z), where σ denotes the nontrivial element of Σ. Identifying σ with the matrix [ 0 11 0 ], the actions are compatible (Assumption 1) if we let Σ act on G by conjugation: σ ·A = σAσ−1. Note that the actions of G and Σ on P are symplectic. The Lie algebra u(2) of U(2) is identified with the skew hermitian matrices, which we write as ξ = [ iu β −β¯ iv ] where u and v are real and β ∈ C. The equivariant momentum map for the U(2) action is given by 〈J(z, w), ξ〉 = 1 2 Ω ( ξ [ z w ] , [ z w ]) = u|z|2 + v|w|2 + 2Im(βz¯w). Assumption 2 reads 〈J(σ(z, w)), ξ〉 = 〈J(z, w), σξσ−1〉, which is easily checked. In this example, GΣ consists of elements of G that commute with σ; that is, GΣ is the subgroup of U(2) consisting of matrices of the form [ a b b a ] . Note that PΣ = { (z, z) ∈ P | z ∈ C } and that the GΣ action on PΣ is (z, z) 7→ ((a + b)z, (a + b)z). We identify the Lie algebra of GΣ with gΣ consisting of matrices of the form ξ = i [ u yy u ] where u and y are real. Note that 〈J(z, z), ξ〉 = 2(u+ y)|z|2, consistent with Proposition 8.1.6. In §8.5, we shall see an alternative way of viewing this example in terms of dual pairs using SU(2) and S1 separately, rather than as U(2). This will also bring out links with the one-to-one resonance more clearly. The next example is an elementary physical example in which we deal with cotangent bundles. B. We consider the spherical pendulum with P = T ∗Q where Q = S2l , the two sphere of radius l. Here G = S1 acts on Q by rotations about the z-axis so that the corresponding momentum map is simply the angular 162 8. Discrete Reduction momentum about the z-axis (with the standard identifications). We let Σ = Z2 act on S2l by reflection in a chosen plane, say the yz-plane, so σ(z, y, z) = (−x, y, z), where q = (x, y, z) ∈ Q. This action is lifted to P by cotangent lift, so it is symplectic. We let σ act on G by conjugation; if Rθ is the rotation about the z-axis through an angle θ, then σ ·Rθ = σRθσ−1 = R−θ. Note that the group ΣsG is O(2). To check Assumptions 1 and 2, it suffices to check Assumption 1Q. It states that σ(Rθ · q) = (σRθσ−1)(σ(q)) which is obviously correct. Here PΣ = T ∗(QΣ) where QΣ = S1l is S 2 l inter- sect the yz-plane. Restriction to T ∗(QΣ) gives a simple pendulum moving in the yz-plane. Here GΣ = {e,Rpi}, so the discrete reduced space is PΣ/Z2, a symplectic orbifold (see Sjamaar [1990]). Thus, we essentially find that discrete reduction of the spherical pendulum is the simple pendulum mov- ing in the yz-plane. Note that here one has different choices of Σ that correspond to different planes of swing of the simple pendulum. Note also, consistent with this, that the angular momentum vanishes on PΣ. C. Here we consider the double spherical pendulum . Let P = T ∗Q where Q = S2l1 × S2l2 and (q1,q2) ∈ Q gives the configuration of the two arms of the pendulum. Again, we let G = S1 consist of rotations about the z-axis, with G acting by the diagonal action on Q. We again let Σ = Z2, and let σ act by reflection in the yz-plane, acting simultaneously on both factors in Q = S2 × S2. As in Example B, this leads via discrete reduction from the spherical double pendulum to the planar double pendulum and GΣ = {e,Rpi}. Another way Z2 acts on Q that is useful in our study of relative equilibria is as follows. Let σ ∈ Z2 be the nontrivial element of Z2 and let σ map (q1,q2) to (q1, σpq2), where P is the vertical plane spanned by k and q1 and where σP is the reflection in this plane. This time, Σ acts trivially on G, so ΣsG is the direct product . Compatible with the general theory in the next section, the subblocking property associated with this symmetry is what gave the subblocking property we observed in equation (5.5.4) in which we calculated δ2Vµ for purposes of the stability analysis. Notice that interchanging the two pendula does not produce a discrete symmetry because the presence of gravity leads to a noninvariance of the kinetic and potential energies under particle interchange. An example with a slightly richer discrete symmetry group is the system of two whirling masses. 8.3 Examples 163 D. Here we let P = T ∗Q where Q = R3 × R3 and (q1,q2) ∈ Q give the configuration of the two masses of the system, as in Figure 8.3.1. The masses are connected to each other and to the supports by springs. The Lagrangian is the standard one: kinetic minus potential energy. However, in this example, the potential energy comes only from the springs; gravity is ignored. Figure 8.3.1. Two whirling masses. Let G = S1 act by rotations of both masses about the (vertical) z-axis. The momentum map is the angular momentum about the vertical axis. We again let Σ = Z2, but now there are two cases: Case 1. Here let σ act by reflection in a vertical plane, say the yz-plane, acting simultaneously on both factors in Q. As above, this action produces an associated invariant subsystem on its fixed point space, the problem of two planar masses connected by springs. This is the case even if the masses of the two particles are different and if the springs are different. As in Example B, the group ΣsG is O(2). As with the double spherical pendulum we can consider the discrete symmetry of reflection in a moving vertical plane. This is important again in the study of relative equilibria. Case 2. Here we let the masses of the two particles be the same, the two outside springs be identical and all three springs be reflection invariant. Now let Σ act on Q by simultaneous reflection in the horizontal plane (and translation of the vector base points to the opposite support). In this case the actions of G and S on Q (and hence on P ) commute, so we can let Σ act trivially on G and Assumption 1Q holds. Here, ΣsG = Z2 × SO(2), 164 8. Discrete Reduction a direct product. Now GΣ = S1 and PΣ = T ∗R3 corresponds to identical motions of the two masses. Thus, in this case discrete reduction gives the motion of a symmetric two mass system on which S1 still acts. This richer collection of discrete symmetries then naturally leads to a rich block structure in the second variation δ2Vµ and a simpler normal form for the linearized equations at a relative equilibrium. See Zombro and Holmes [1993] for more information. E. The rigid body has three copies of Z2 as symmetry group. Here Q = SO(3) and we choose a plane P corresponding to one of the principle moments of inertia and let σ be the reflection in this plane. Let Σ = Z2 be generated by σ. Let P = T ∗Q, and G = SO(3). Let G act on Q by left multiplication (as is usual for the rigid body), so the momentum map is the spatial angular momentum of the body. The discrete symmetries in this example yield the well known symmetry of the phase portrait of the rigid body as viewed on the two sphere. Let Σ act on Q by conjugation; for A ∈ Q, σ ·A = σAσ−1. If we let Σ act on G the same way, then σQ(RQ(A)) = σRAσ−1 = (σRσ−1)(σAσ−1) = σG(R) · σQ(A), so Assumption 1Q holds. As in Example B, the semidirect product is the orthogonal group: ΣsG = O(3). In this case, GΣ consists of rotations in the plane defining σ and PΣ = T ∗GΣ. Here, the discrete reduction yields a rigid rotor constrained to rotate about a fixed principal axis. Remark This example illustrates two cautions that are needed when dealing with discrete symplectic symmetries. First, while there is a well defined action of Σ on P/G, this action need not be Poisson. For example the induced action on so(3)∗ in Example E is anti-Poisson. Also, for σ ∈ Σ, σ maps J−1(µ) to J−1(σg∗µ), so there need not be a well defined action on Pµ, let alone a symplectic one. (For the rigid body one gets an anti- symplectic map of S2 to itself.) F. We modify Example E to a situation that is of interest in pseudo rigid bodies and gravitating fluid masses (see Lewis and Simo [1990]). Let Q = GL(3) (representing linear, but nonrigid deformations of a reference configuration). Let G = SO(3) act on the left on Q as before, so again, the momentum map represents the total spatial angular momentum of the system. As in Example E, let Σ = Z2 and let σ be reflection in a chosen, fixed plane P. Let σS denote reflection in the image of the plane P under linear transformation S, and let the action of σ on an element A of Q be on the right by σ and on the left by σA and let it act trivially on G. Again, one checks that Assumption 1Q holds. In this case, the semidirect product is the direct product Z2 × SO(3), so the full rotation group still acts on PΣ =. Now, QΣ represents configurations that have a plane of symmetry, so in this case, discrete reduction of this system corresponds to restriction to symmetric bodies. 8.4 Sub-Block Diagonalization with Discrete Symmetry 165 In view of the block diagonalization results outlined in the next section, it is of interest to not to actually carry out the discrete reduction, but rather to use the discrete symmetry to study stability. This seems to explain, in part, why, in their calculations, further subblocking, in some cases, even to diagonal matrices, was found. 8.4 Sub-Block Diagonalization with Discrete Symmetry In this section, we explain how Lemma 8.1.5 gives a subblocking theorem in the energy–momentum method. We indicate how this result applies to the classical water molecule without carrying out the calculations in detail here. In this procedure, we do not carry out discrete reduction explicitly (although it does give an interesting invariant subsystem for the dynam- ics of symmetric molecules). Rather, we use it to divide the modes into symmetric and non-symmetric ones. Recall from Chapter 5, that one splits the space of variations of the concrete realization V of Q/Gµ into variations VRIG in G/Gµ and variations VINT in Q/Gµ. With the appropriate splitting, one gets the block diagonal structure δ2Vµ =  Arnold form 0 0 Smale form  where the Arnold form means δ2Vµ computed on the coadjoint orbit tangent space, and the Smale form means δ2Vµ computed on Q/G. Perhaps even more interesting is the structure of the linearized dynamics near a relative equilibrium. That is, both the augmented Hamiltonian Hξ = H−〈J, ξ〉 and the symplectic structure can be simultaneously brought into the following normal form: δ2Hξ =  Arnold form 0 0 0 Smale form 0 0 0 Kinetic Energy > 0  and Symplectic form =  coadjoint orbit form C 0 −C magnetic (coriolis) I 0 −I 0  166 8. Discrete Reduction where the columns represent the coadjoint orbit variables (G/Gµ), the shape variables (Q/G) and the shape momenta respectively. The term C is an interaction term between the group variables and the shape variables. The magnetic term is the curvature of the µ-component of the mechanical connection, as we described earlier. Suppose that we have a compact discrete group Σ acting by isometries on Q, and preserving the potential, so we are in the setting of the preceding section. This action lifts to the cotangent bundle, as we have seen. The resulting fixed point space is the cotangent bundle of the fixed point space QΣ. This fixed point space represents the Σ-symmetric configurations. Of more concern here is the fact that the action also gives an action on the quotient space, or shape space Q/G. We can split the tangent space to Q/G at a configuration corresponding to the relative equilibrium according to Lemma 8.1.4 and can apply Lemma 8.1.5 to the Smale form. Here, one must check that the amended potential is actually invariant under Σ. In general, this need not be the case, since the discrete group need not leave the value of the momentum µ invariant. However, there are two important cases for which this is verified. The first is for SO(3) with Z2 acting by conjugation, as in the rigid body example, it maps µ to its negative, so in this case, from the formula Vµ(q) = V (q)−µI(q)−1µ we see that indeed Vµ is invariant. The second case, which is relevant for the water molecule, is when Σ acts trivially on G. Then Σ leaves µ invariant, and so Vµ is again invariant. 8.4.1 Theorem. Under these assumptions, at a relative equilibrium, the second variation δ2V (q) block diagonalizes, which we refer to as the sub- blocking property . The blocks in the Smale form are given by Lemma 8.1.4: they are the Σ-symmetric variations, and their complement chosen according to that lemma as the annihilator of the symmetric dual variations. Lemma 8.1.5 can also be applied to the symplectic form, showing that it subblocks as well. These remarks apply to the classical rotating water molecule as follows. (We let the reader work out the case of the two whirling masses in a similar, but simpler vein.) The discrete symmetry Σ is Z2 acting, roughly speak- ing, by interchanging the two hydrogen atoms. We shall describe the action more precisely in a moment. The action of Z2 on SO(3) will be the trivial one, so the semi-direct product is the direct product and so GΣ is again SO(3). The discrete reduction procedure then yields the system consisting of symmetric molecules (with the two hydrogen atoms moving in a symmet- ric way), but still with symmetry group the rotation group, which makes good sense physically. The subblocking property mentioned above shows that the Smale form, which in this example, is a 3 × 3 symmetric matrix, becomes block diagonal, with a 2 × 2 subblock corresponding to the sym- metric variations, and a singleton block corresponding to the nonsymmetric 8.4 Sub-Block Diagonalization with Discrete Symmetry 167 variations. Explicitly, the nontrivial element σ of Σ acts on the configuration space as follows: σ(r, s) = ( r, s− 2(r · s) r‖ r ‖2 ) . This symmetry interchanges the role of the two masses as in Figure 8.4.1. m m M M m m Figure 8.4.1. Discrete symmetry of the water molecule. If one prefers, σ is the transformation that rotates the system by 180 degrees about a line perpendicular to r in the plane of the system, followed by an interchange of the particles. One now checks the claims made ear- lier; for example, that the Assumption 1Q is satisfied, and that the fixed point set of the discrete symmetry group does correspond to the symmetric configurations, etc. Notice that for the ozone molecule, one can apply this operation to any two pairs of atoms, and this leads to interesting conse- quences such as, for the planar molecule, the “breathing mode” decouples dynamically from the other modes, and in space, it couples very simply via the internal rigid coupling discussed in Chapter 5. We conclude that δ2Vµ takes the following form: δ2Vµ =  a b 0b d 0 0 0 g  (8.4.1) where the block [ a b b d ] corresponds to the symmetric variations and [g] to the symmetry breaking variations. Another system in which an interesting discrete symmetry occurs is the rotating liquid drop, as studied by Lewis, Marsden, and Ratiu [1987] and Lewis [1989]. Here, one has an ideal incompressible, inviscid fluid system (so 168 8. Discrete Reduction the system is infinite dimensional) that has a basic symmetry group S1 for a planar drop, and SO(3) for a drop in space. In either case, the corresponding momentum map is the angular momentum of the drop. However, there is a discrete symmetry as well, that can be set up in a way somewhat similar to that of the rigid body, described in §8.3. For the planar case, one chooses a line say L in the reference configuration of the fluid and calls the reflection in L by σL. The configuration space for the drop is Q = all volume preserving embeddings of a reference configuration to the plane. Then σL acts on an element η ∈ Q by conjugation, on the right by σL and on the left by the reflection in the line that is the line through the images η(P1), η(P2), where P1, P2 are the intersection points of the line with the boundary of the reference configuration. This action is then lifted to the cotangent space in the standard way. Here the fixed point set is the set of symmetric drops, and the action of the discrete group on the group S1 is trivial, so the fixed point group GΣ is again S1 and it still acts on the symmetric drops, as it should. One can view the planar water molecule as a finite dimensional analogue of this model. These discrete symmetries are clearly available for other problems as well, such as the classical problem of rotating gravitational fluid masses. In this case, one has a richer symmetry structure coming from finite subgroups of the orthogonal group O(3). From the general discussion above, the sub- blocking property that the discrete symmetry gives, should be useful for the study of stability and bifurcation of these systems. 8.5 Discrete Reduction of Dual Pairs We now give a set up one can use in the application of the discrete reduction procedure to integrable Hamiltonian systems. Consider two Lie groups G andH acting symplectically on the manifold P , and assume that the actions commute. If g and h denote the Lie algebras of G and H respectively, we suppose that there are equivariant momentum maps JG : P → g∗ and JH : P → h∗ (8.5.1) generating the actions. Dual Pair Assumption. For all z ∈ P : J−1G (JG(z)) = OHz and J−1H (JH(z)) = OGz (8.5.2) where OHz and OGz denote the orbits of H and G respectively through z. We also assume that the group Σ acts on P , G, and H and hence on g, h and g∗ and h∗ where Assumptions 1 and 2 of §8.1 hold for both G and H (Figures 8.5.1 and 8.5.2). 8.5 Discrete Reduction of Dual Pairs 169 P P P P gP or hP [σG(g)]P or [σH(h)]P σP σP - - ? ? Figure 8.5.1. Assumption 1 for dual pairs. g∗ff JG JG P - JH JH h∗ g∗ff P - h∗ ? σg∗ σP ? σh∗ ? Figure 8.5.2. Assumption 2 for dual pairs. Assuming also that the action of H is free and proper, so that P → P/H is an H-fibration, it follows that the mapping induced by J JG : P/H → g∗ is a Poisson embedding, where g∗ has the Lie–Poisson structure (the minus structure if the actions are left, and the plus structure if the actions are on the right) and where P/H has the quotient Poisson structure. Thus, we have the commutative diagram in Figure 8.5.3. g∗Σ h ∗ Σ JGΣ JHΣ PΣ ff - J¯GΣ proj proj J¯HΣ PΣ/HΣ PΣ/GΣ @ @ @ @I @ @ @ @R Figure 8.5.3. Reduction of dual pairs. For the following proposition, note that OGΣz ⊂ OGz ∩ PΣ and similarly for H. 8.5.1 Proposition. Let (P,G,H) be a dual pair, acted on, as above, by Σ. Suppose that for each z ∈ PΣ, OGΣz = OGz ∩ PΣ and OHΣz = OHz ∩ PΣ. (8.5.3) Then (PΣ, GΣ, HΣ) form a dual pair. 170 8. Discrete Reduction Proof. That PΣ is a symplectic manifold, with commuting Hamilto- nian actions of GΣ, HΣ follows from Propositions 8.1.1 and 8.1.2. Proposi- tion 8.1.6 then implies that one has momentum maps JGΣ ,JHΣ given by restrictions of JG and JH . The conditions (8.5.3) guarantee that the dual pair assumption is satisfied. Let us consider briefly the condition (8.5.3). As we noted, we always have the inclusion OHΣz ⊂ OHz ∩ PΣ. Let z′ = hz belong to OHz ∩ PΣ. As σP (z′) = z′, and σP (z) = z, for all σ ∈ Σ, one obtains: h−1σH(h) ∈ StabH(z). If σ is an involution, h−1σH(h) belongs to the subset Stab−H(z) of elements t satisfying σ(t) = t−1. One wants an hˆ = hs−1, where s ∈ StabH(z), such that σH(hˆ) = hˆ, for all σ ∈ Σ. One must then solve the equations: h−1σ(h) = s−1σ(s) for all σ ∈ Σ for some s ∈ StabH(z). This is frequently possible: one simple case is when the action of H is free. When Σ is generated by an involution σ, (i.e., Σ = Z2), the preceding equation will be solvable if the map Πσ : StabH(z)→ Stab−H(z); s 7→ s−1σ(s) is surjective. If the action of H (and so of HΣ) is free and proper and JG is a submer- sion along J−1G (g ∗ Σ) (or at least over some open subset of g ∗ Σ), then one has an H-bundle in P JG : J−1G (g ∗ Σ)→ J−1G (g∗Σ)/H ⊂ g∗Σ (8.5.4) and an HΣ bundle in PΣ JGΣ : J −1 GΣ : (g∗Σ)→ J−1GΣ(g∗Σ)/HΣ ⊂ g∗Σ (8.5.5) giving an inclusion: J−1GΣ(g ∗ Σ)/HΣ ⊂ J−1G (g∗Σ)/H. Over J−1GΣ(g ∗ Σ)/HΣ, (8.5.5) is a subbundle of (8.5.4). This, in fact, occurs over an open set: 8.5.2 Proposition. Under the hypotheses given above, J−1GΣ(g ∗ Σ)/HΣ is an open subset of J−1G (g ∗ Σ)/H. Proof. The set J−1GΣ(g ∗ Σ)/HΣ consists of those points ξ in g ∗ Σ for which there is a z in PΣ with JGΣ(z) = JG(z) = ξ. By Corollary 8.1.8, this set is open in J−1G (g ∗ Σ)/H. 8.5 Discrete Reduction of Dual Pairs 171 Example. Consider the situation of Example A of §8.3. We choose the same P = C2 and the same symplectic structure. However, now we choose G = SU(2) and H = U(1) = S1 with G acting by matrix multiplication and with H acting by the flow of two identical harmonic oscillators, with Hamiltonian given by H(z, w) = 12 (|z|2 + |w|2). These form a dual pair with the momentum maps given by the components of the Hopf map; see Cushman and Rod [1982] and Marsden [1987]. Again, we choose the discrete group Σ to be Z2 given by the map σ(z, w) = (w, z). Here the hypotheses above are directly verified, and one finds, using the sort of computations in Example A of §8.3, that the discrete reduction of this dual pair is the trivial dual pair given by PΣ = C2 with GΣ = S1 acting by complex multiplication and with HΣ = S1 acting the same way. In other words, the discrete reduction of the (completely integrable) system defined by the one- to-one resonance consisting of two identical harmonic oscillators (integrable via the group SU(2)), is the (completely integrable) system consisting of a single harmonic oscillator, trivially integrable using the group S1. For more sophisticated examples of integrable systems, and in particular, the Kowalewski top, see Bobenko, Reyman, and Semenov-Tian-Shansky [1989]. 172 8. Discrete Reduction Page 173 9 Mechanical Integrators For conservative mechanical systems with symmetry, it is of interest to develop numerical schemes that preserve this symmetry, so that the associ- ated conserved quantities are preserved exactly by the integration process. One would also like the algorithm to preserve either the Hamiltonian or the symplectic structure—one cannot expect to do both in general, as we shall show below. There is some evidence (such as reported by Channell and Scovel [1990], Marsden, O’Reilly, Wicklin, and Zombro [1991] and Pullin and Saffman [1991]) that these mechanical integrators perform especially well for long time integrations, in which chaotic dynamics can be expected. Some standard algorithms can introduce spurious effects (such as nonex- istent chaos) in long integration runs; see, for example, Reinhall, Caughy, and Storti [1989]. We use the general term mechanical integrator for an algorithm that respects one or more of the fundamental properties of being symplectic, preserving energy, or preserving the momentum map. 9.1 Definitions and Examples By an algorithm on a phase space P we mean a collection of maps Fτ : P → P (depending smoothly, say, on τ ∈ R for small τ and z ∈ P ). Sometimes we write zk+1 = Fτ (zk) for the algorithm and we write ∆t or h for the step size τ . We say that the algorithm is consistent or is first 174 9. Mechanical Integrators order accurate with a vector field X on P if d dτ Fτ (z) ∣∣∣∣ τ=0 = X(z). (9.1.1) Higher order accuracy is defined similarly by matching higher order deriva- tives. One of the basic things one is interested in is convergence namely, when is lim n→∞(Ft/n) n(z) = ϕt(z) (9.1.2) where ϕt is the flow of X, and what are the error estimates? There are some general theorems guaranteeing this, with an important hypothesis being stability ; that is, (Ft/n)n(z) must remain close to z for small t and all n = 1, 2, . . .. We refer to Chorin, Hughes, Marsden, and McCracken [1978] and Abraham, Marsden, and Ratiu [1988] for details. For example, the Lie–Trotter formula et(A+B) = lim n→∞(e tA/netB/n)n (9.1.3) is an instance of this. An algorithm Fτ is 1. a symplectic-integrator if each Fτ is symplectic, 2. an energy-integrator if H ◦ Fτ = H (where X = XH), 3. a momentum-integrator if J ◦ Fτ = J (where J is the momentum map for a G-action). If Fτ has one or more of these properties, we call it a mechanical inte- grator . Notice that if an integrator has one of these three properties, then so does any iterate of it. There are two ways that have been employed to find mechanical inte- grators. For example, one can search amongst existing algorithms and find ones with special algebraic properties that make them symplectic or energy- preserving. Second, one can attempt to design mechanical integrators from scratch. Here are some simple examples. A. A first order explicit symplectic scheme in the plane is given by the map (q0, p0) 7→ (q, p) defined by q = q0 + (∆t)p0 p = p0 − (∆t)V ′(q0 + (∆t)p0). (9.1.4) This map is a first order approximation to the flow of Hamilton’s equations for the Hamiltonian H = (p2/2) + V (q). Here, one can verify by direct calculation that this scheme is in fact a symplectic map. 9.1 Definitions and Examples 175 Example A is based on the use of generating functions, as we shall see below. A modification of Example A using Poincare´’s generating function, but also one that can be checked directly is: B. An implicit symplectic scheme in the plane for the same Hamiltonian as in Example A is q = q0 + (∆t) p+ p0 2 p = p0 − (∆t)V ′(q + q02 ). (9.1.5) Other examples are sometimes based on special observations. The next example shows that the second order accurate mid-point rule is symplectic ( Feng [1986]). This algorithm is also useful in developing almost Poisson integrators (Austin, Krishnaprasad, and Wang [1993]). C. In a symplectic vector space the following mid point rule is sym- plectic: zk+1 − zk ∆t = XH ( zk + zk+1 2 ) . (9.1.6) Notice that for small ∆t the map defined implicitly by this equation is well defined by the implicit function theorem. To show it is symplectic, we use the fact that the Cayley transform S of an infinitesimally symplectic linear map A, namely S = (1− λA)−1 (1 + λA) (9.1.7) is symplectic if 1 − λA is invertible for some real λ. To apply this to our situation, rewrite the algorithm (9.1.6) as Fτ (z)− z − τXH ( z + Fτ (z) 2 ) = 0. (9.1.8) Letting S = dFτ (z) and A = dXH ( z+Fτ (z) 2 ) we get, by differentiation, S − 1− 12τA(1 + S) = 0; that is, (9.1.7) holds with λ = τ/2. Thus, (9.1.6) defines a symplectic scheme. D. Here is an example of an implicit energy preserving algorithm from Chorin, Hughes, Marsden, and McCracken [1978]. Consider a Hamiltonian system for q ∈ Rn and p ∈ Rn: q˙ = ∂H ∂p , p˙ = −∂H ∂q . (9.1.9) 176 9. Mechanical Integrators Define the following implicit scheme qn+1 = qn + ∆t H(qn+1,pn+1)−H(qn+1,pn) λT (pn+1 − pn) λ, (9.1.10) pn+1 = pn −∆tH(qn+1,pn)−H(qn,pn) µT (qn+1 − qn) µ, (9.1.11) where λ = ∂H ∂p (αqn+1 + (1− α)qn, βpn+1 + (1− β)pn), (9.1.12) µ = ∂H ∂q (γqn+1 + (1− γ)qn, δpn+1 + (1− δ)pn), (9.1.13) and where α, β, γ, δ are arbitrarily chosen constants in [0, 1]. The proof of conservation of energy is simple: From (9.1.10), we have (qn+1 − qn)T (pn+1 − pn) = ∆t(H(qn+1,pn+1)−H(qn+1,pn)), (9.1.14) and from (9.1.11) (pn+1 − pn)T (qn+1 − qn) = −∆t(H(qn+1,pn)−H(qn,pn)). (9.1.15) Subtracting (9.1.15) from (9.1.14), we obtain H(qn+1,pn+1) = H(qn,pn). (9.1.16) This algorithm is checked to be consistent. In general, it is not symplectic— this is in accord with Proposition 9.2.1 in the next section. E. Let us apply the Lie–Trotter or time splitting idea to the simple pendulum. The equations are d dt ( ϕ p ) = ( p 0 ) + ( 0 − sinϕ ) . Each vector field can be integrated explicitly to give maps Gτ (ϕ, p) = (ϕ+ τp, p) and Hτ (ϕ, p) = (ϕ, p− τ sinϕ) each of which is symplectic. Thus, the composition Fτ = Gτ ◦Hτ , namely, Fτ (q, p) = (ϕ+ τp− τ2 sinϕ, p− τ sinϕ) is a first order symplectic scheme for the simple pendulum. It is closely related to the standard map. The orbits of Fτ need not preserve energy and they may be chaotic, whereas the trajectories of the simple pendulum are of course not chaotic. We refer to the cited references, and to Ruth [1983], Feng [1986], Sanz- Serna [1988] and references therein for more examples of this type, including symplectic Runge–Kutta schemes. 9.2 Limitations on Mechanical Integrators 177 Variational Methods. A relatively new and powerful method for gener- ating symplectic-momentum integrators is the variational technique, which is described in Wendlandt and Marsden [1997]; see also MacKay [1992]. These are schemes that are based on a discrete version of Hamilton’s prin- ciple and automatically lead to symplectic momentum integrators. The discrete Lagrangian is in effect, the generating function for the algorithm in the sense of §9.3 below. In Simo, Tarnow, and Wong [1992], it is noted that “All of the im- plicit members of the Newmark family, perhaps the most widely used time- stepping algorithms in nonlinear structural dynamics, are not designed to conserve energy and also fail to conserve momentum. Among the explicit members, only the central difference method preserves momentum. Despite this remark, it is shown that the Newmark family in fact is symplectic in Kane, Marsden, Ortiz, and West [1999]. This is done by showing that the algorithm is in fact, variational. It does not contradict the above quote be- cause the conserved symplectic form differs from the “obvious” one, which is not strictly conserved, by higher order terms in the time step, so is algo- rithmically just as good a choice. 9.2 Limitations on Mechanical Integrators A number of algorithms have been developed specifically for integrating Hamiltonian systems to conserve the energy integral, but without attempt- ing to capture all of the details of the Hamiltonian structure (see Example D above and also Stofer [1987] and Greenspan [1974, 1984]). In fact, some of the standard energy-conservative algorithms have poor momentum be- havior over even moderate time ranges. This makes them unsuitable for problems in satellite dynamics for example, where the exact conservation of a momentum integral is essential to the control mechanism. One can get angular momentum drift in energy-conservative simulations of, for example, rods that are free to vibrate and rotate, and presumably in the water molecule. To control such drifts and attain the high levels of computational accuracy demanded by automated control mechanisms, one would be forced to reduce computational step sizes to such an extent that the numerical simulation would be prohibitively inefficient. Similarly, if one attempts to use a standard energy-conservative algorithm to simulate both the rotational and vibrational modes of a freely moving flexible rod, the algorithm may predict that the rotational motion will come to a virtual halt after only a few cycles! For a documented simulation of a problem with momentum conservation, see Simo and Wong [1989]. One can readily imagine that in the process of enforcing energy conservation one could upset conservation of angular momentum. 178 9. Mechanical Integrators As we shall demonstrate in §9.4, the problem of how to generate a nu- merical algorithm that exactly conserves momentum (or, more generally, all momentum-like integrals of motion) is fairly easy to resolve. Since mo- mentum integrals in Hamiltonian systems are associated with invariance of the system under the action of symmetry groups, one might guess that to derive momentum-conservative algorithms, one constrains the algorithm to obey, in some sense, the same group invariance as the actual dynamics. In traditional integrators, much attention has been paid to energy con- servation properties, some, as we have noted to momentum conservation, and even less to conserving the symplectic or Poisson structure. However, one can imagine that it is also quite important. The three notions of symplectic, energy, and momentum integrators are connected in interesting ways. For example, as we shall show below, a re- sult of Ge (see Ge and Marsden [1988]) is that under fairly weak additional assumptions, a G-equivariant symplectic integrator is also momentum pre- serving. For example, a symplectic integrator of this type applied to a free rigid body motion would exactly preserve the angular momentum vector in space. Given the importance of conserving integrals of motion and the impor- tant role played by the Hamiltonian structure in the reduction procedure for a system with symmetry, one might hope to find an algorithm that combines all of the desirable properties: conservation of energy, conserva- tion of momenta (and other independent integrals), and conservation of the symplectic structure. However, one cannot do all three of these things at once unless one relaxes one or more of the conditions in the following sense: 9.2.1 Proposition. If an algorithm is energy preserving, symplectic and momentum preserving and if the dynamics is nonintegrable on the reduced space (in the sense spelled out in the proof ) then the algorithm already gives the exact solution of the original dynamics problem up to a time reparametrization. Proof. Suppose F∆t is our symplectic algorithm of the type discussed above, and consider the application of the algorithm to the reduced phase space. We assume that the Hamiltonian H is the only integral of motion of the reduced dynamics, that is, all other integrals of the system have been found and taken out in the reduction process in the sense that any other conserved quantity (in a suitable class) is functionally dependent on H. Since F∆t is symplectic it is the ∆t-time map of some time-dependent Hamiltonian function K. Now assume that the symplectic map F∆t also conserves H for all values of ∆t. Thus {H,K} = 0 = {K,H}. The latter equation implies that K is functionally dependent on H since the flow of H (the “true dynamics”) had no other integrals of motion. The functional dependence of K on H in turn implies that their Hamiltonian vector fields are parallel, so the flow of K (the approximate solution) and the flow of H 9.3 Symplectic Integrators and Generating Functions 179 (the exact solution) must lie along identical curves in the reduced phases space; thus the flows are equivalent up to time reparametrization. Thus, it is unlikely one can find an algorithm that simultaneously con- serves the symplectic structure, the momentum map, and the Hamiltonian. It is tempting (but probably wrong) to guess from this that one can moni- tor accuracy by keeping track of all three. Non-symplectic algorithms that conserve both momentum and energy have been studied by Simo and Wong [1989] and Austin, Krishnaprasad, and Wang [1993]. We study the basic method in §9.5. It is interesting to note that when adaptive time steps are used, the arguments above no longer apply and indeed in this case it is possible to find integrators that are, in an appropriate sense, symplectic, energy preserving and momentum preserving, as shown in Kane, Marsden, and Ortiz [1999]. 9.3 Symplectic Integrators and Generating Functions Symplectic integrators based on generating functions have been developed by a large number of people, starting with de Vogelae´re [1956] and Feng [1986]. We refer to Channell and Scovel [1990] and Ge [1991] for a survey. Let us recall the following basic fact. 9.3.1 Proposition. If S : Q×Q→ R defines a diffeomorphism (q0, p0) 7→ (q, p) implicitly by p = ∂S ∂q and po = − ∂S ∂q0 (9.3.1) then this diffeomorphism is symplectic. Proof. Note that dS = ∂S ∂qi dqi + ∂S ∂q io dq io = pidq i − poidq io and so taking d of both sides, we get dpi ∧ dqi = dpoi ∧ dq io which means the diffeomorphism (9.3.1) is symplectic. Recall also one of the basic facts about Hamilton–Jacobi theory, namely that the flow of Hamilton’s equations is the canonical transformation gen- erated by the solution of the Hamilton–Jacobi equation ∂S ∂t +H ( q, ∂S ∂q ) = 0 (9.3.2) 180 9. Mechanical Integrators where S(q0, q, t)|q=q0 t=0 generates the identity. (This may require singular be- havior in t; for example, consider S = 12t (q − q0)2.) The strategy is to find an approximate solution of the Hamilton–Jacobi equation for small time ∆t and to use this to obtain the algorithm using (9.3.1). There are several other versions of the algorithm that one can also treat. For example, if specific coordinates are chosen on the phase space, one can use a generating function of the form S(qi, p0i, t). In this case one can get the simple formula for a first order algorithm given in Example A in §9.1 by using S = p0iqi − ∆tH(qi, p0i), which is easy to implement, and for Hamiltonians of the form kinetic plus potential, leads to the stated explicit symplectic algorithm. As explained in Ge [1991], one can use other types of generating functions. For example, using the Poincare´ generating function, one recovers the mid-point scheme. 9.4 Symmetric Symplectic Algorithms Conserve J The construction of momentum-conserving algorithms, whether of symplec- tic or energy–momentum type, requires that level sets of the momentum map J remain invariant under the mapping ϕ : P → P that represents a single iteration of the algorithm. We next give sufficient conditions under which it is possible to obtain such a mapping in the symplectic case. The argument is a modification of some ideas found in Ge and Marsden [1988]. See also Ge [1991] and references therein. We make the following assumptions: (i) G is a Lie group acting symplectically on P and J : P → g∗ is an associated momentum map for the action, with g representing the action of g ∈ G; (ii) ϕ : P → P is a symplectic map; (iii) ϕ is G-equivariant: ϕ(gz) = gϕ(z), for all z ∈ P and g ∈ G. Letting ξP = X〈J,ξ〉 designate the vector field corresponding to ξ ∈ g under the action, we start by differentiating the equivariance condition ϕ ◦ g = g ◦ ϕ with respect to the group element in the direction of ξ at the identity of the group. This results in ϕ∗ξP = ξP or, by definition of the momentum map, ϕ∗X〈J,ξ〉 = X〈J,ξ〉. But ϕ∗X〈J,ξ〉 = X〈J,ξ〉◦ϕ from Assumption (iii), so X〈J,ξ〉◦ϕ = X〈J,ξ〉. 9.4 Symmetric Symplectic Algorithms Conserve J 181 Since two Hamiltonian vector fields are equal if and only if the Hamiltonians differ by a constant, 〈J, ξ〉 ◦ ϕ− 〈J, ξ〉 = constant. We need 〈J, ξ〉 ◦ϕ = 〈J, ξ〉 for the value of J to be preserved by the map ϕ, so we need to establish sufficient conditions under which the constant will vanish. An example of such conditions are: (iv) P is a symplectic manifold endowed with an exact symplectic form ω = −dθ; (v) ϕ∗θ − θ, which is closed, is also G− exact; that is, if ϕ∗θ − θ = dS, then S : P → R is a G-invariant generating function for the map ϕ; (vi) 〈J, ξ〉 = iξpθ. Then, 〈J, ξ〉 ◦ ϕ = ϕ∗〈J, ξ〉 = ϕ∗iξpθ (by (vi)) = iξpϕ∗θ (by equivariance of ϕ) = iξpθ + iξpdS (by (v)). The first term in this last expression is 〈J, ξ〉 again, and the final term vanishes by equivariance of S. Thus, the desired conservation condition, 〈J, ξ〉 ◦ ϕ = 〈J, ξ〉, follows from these assumptions. Assuming that the original system is given in terms of canonical coordi- nates on a cotangent bundle P = T ∗Q, we have ω = −dθ0, where θ0 = pidqi is the canonical one-form on the cotangent bundle. If the symmetry group G acts by cotangent lifts, then vi follows automatically. Condition v is equivalent to (9.1.1) if we regard S as a function of q0 and q. This argument applies to all “types” of generating functions, but when applied to ones of the form S(q0, q, t) we get: 9.4.1 Proposition. Suppose that S : Q × Q → R is invariant under the diagonal action of G, that is, S(gq, gq0) = S(q, q0). Then the cotan- gent momentum map J is invariant under the canonical transformation ϕS generated by S, that is, J ◦ ϕS = J. This may also be seen directly by differentiating the invariance condition assumed on S with respect to g ∈ G in the direction of ξ ∈ g and utilizing the definitions of ϕS , J, and ξQ. The following is also true: If G acts on Q freely, and a given canonical transformation ϕ conserves J, then its generating function S can be defined on an open set of Q × Q which is invariant under the action of G, and S is invariant under the action of G. This is proved in Ge [1991]. 182 9. Mechanical Integrators Note that if H is invariant under the action of G, then the correspond- ing solution of the Hamilton–Jacobi equation is G invariant as well. This follows from the short time uniqueness of the generating function of the type assumed for the flow of the Hamiltonian vector field XH determined by H. It also follows from Proposition 9.4.1 that if the approximate solu- tion of the Hamilton–Jacobi equation is chosen to be G-invariant, then the corresponding algorithm will exactly conserve the momentum map. For a start on the numerical analysis of symplectic integrators, see Sanz- Serna [1988], Simo, Tarnow, and Wong [1992] and related papers. This whole area needs further development for the community to be able to in- telligently choose amongst various algorithms. For instance, from the point of view of stability, the optimal second-order accurate symplectic integra- tors are the mid point rule and the central difference method. 9.5 Energy–Momentum Algorithms We now turn to some basic remarks on the construction of algorithms that conserves the Hamiltonian and the momentum map, but will not, in general, conserve the symplectic structure. A class of algorithms satisfying this requirement can be obtained through the steps outlined below. The geometry of the process is depicted in Fig- ure 9.5.1. (i) Formulate an energy-preserving algorithm on the symplectic reduced phase space Pµ = J−1(µ)/Gµ or the Poisson reduced space P/G. If such an algorithm is interpreted in terms of the primitive phase space P , it becomes an iterative mapping from one orbit of the group action to another. (ii) In terms of canonical coordinates (q, p) on P , interpret the orbit-to- orbit mapping described above and if P/G was used, impose the con- straint J(qk, pk) = J(qk+1, pk+1). The constraint does not uniquely determine the restricted mapping, so we may obtain a large class if iterative schemes. (iii) To uniquely determine a map from within the above class, we must determine how points in one Gµ-orbit are mapped to points in an- other orbit. There is still an ambiguity about how phase space points drift in the Gµ-orbit directions. This drift is closely connected with geometric phases (Chapter 6)! In fact by discretizing the geometric phase formula for the system under consideration we can specify the shift along each Gµ-orbit associated with each iteration of the map. The papers of Simo and Wong [1989] and Austin, Krishnaprasad, and Wang [1993] provide systematic methods for making the choices required 9.5 Energy–Momentum Algorithms 183 projection reduced space constant energy surfaces -orbits Figure 9.5.1. An energy preserving algorithm is designed on the reduced space and is then lifted to the level set of the momentum map by specifying phase information. in Steps ii and iii. The general construction given above is, in fact, pre- cisely the approach advocated in Simo, Tarnow, and Wong [1992]. There it is shown that projection from the level set of constant angular momentum onto the surface of constant energy can be performed implicitly or explic- itly leading to predictor/corrector type of algorithms. From a numerical analysis standpoint, the nice thing is that the cost involved in the actual construction of the projection reduces to that of a line search (i.e., basi- cally for free). The algorithm advocated in Simo and Wong [1989] is special in the sense that the projection is not needed for Q = SO(3): the discrete flow is shown to lie in the intersection of the level set of angular momentum and the surface of constant energy. This algorithm is singularity-free and integrates the dynamics exactly up to a time reparametrization, consistent with the restrictions on mechanical integrators given in §9.2. Extensions of these schemes to elasticity, rods and shells suitable for large-scale cal- culation and amenable to parallelization are given in Simo, Fox, and Rifai [1991], and Simo and Doblare [1991]. 184 9. Mechanical Integrators 9.6 The Lie–Poisson Hamilton–Jacobi Equation Suppose that we are able to produce a G-invariant generating function S. Since it is G-invariant, it can be reduced, either by symplectic or Poisson reduction to produce an algorithm on the reduced space. It also gives rise to a reduced version of Hamilton–Jacobi theory. This can be applied to, for example, the rigid body in body representation or, in principle, to fluids and plasmas in the spatial representation. Instead of giving the generalities of the theory, we shall illustrate it in an important case, namely, with the case of Lie–Poisson reduction, whereby we take Q = G, so the reduced space T ∗Q/G is isomorphic with the dual space g∗ with the Lie–Poisson bracket (with a plus sign for right reduction and a minus sign for left reduction). We shall give the special case of the rigid body for illustration, taking G = SO(3). Since the momentum map is preserved, one also gets an induced algorithm on the coadjoint orbits, or in the more general cases, on the symplectic reduced spaces. The proofs can be routinely provided by tracing through the definitions. We begin with the reduced Hamilton–Jacobi equation itself. Thus, let H be a G-invariant function on T ∗G and let HL be the corresponding left reduced Hamiltonian on g∗. (To be specific, we deal with left actions—of course there are similar statements for right reduced Hamiltonians.) If S is invariant, there is a unique function SL such that S(g, g0) = SL(g−1g0). (One gets a slightly different representation for S by writing g−10 g in place of g−1g0.) 9.6.1 Proposition. The left reduced Hamilton–Jacobi equation is the following equation for a function SL : G→ R: ∂SL ∂t +HL(−TR∗g · dSL(g)) = 0 (9.6.1) which we call the Lie–Poisson Hamilton–Jacobi equation. The Lie– Poisson flow of the Hamiltonian HL is generated by the solution SL of (9.6.1) in the sense that the flow is given by the Poisson transformation of g∗ : Π0 7→ Π defined as follows: Define g ∈ G by solving the equation Π0 = −TL∗g · dgSL (9.6.2) for g ∈ G and then setting Π = Ad∗g−1Π0. (9.6.3) Here Ad denotes the adjoint action and so the action in (9.6.3) is the coadjoint action. Note that (9.6.3) and (9.6.2) give Π = −TR∗g · dSL(g). Note also that (9.6.2) and (9.6.3) are the analogues of equation (9.1.1) 9.6 The Lie–Poisson Hamilton–Jacobi Equation 185 and that (9.6.1) is the analogue of (9.1.2). Thus, one can obtain a Lie– Poisson integrator by approximately solving (9.6.1) and then using (9.6.2) and (9.6.3) to generate the algorithm. This algorithm (9.6.3) manifestly preserves the coadjoint orbits (the symplectic leaves in this case). As in the canonical case, one can generate algorithms of arbitrary accuracy this way. There may be conditions necessary on Π0 for the solvability of equa- tion (9.6.2). This is noted in the example of the rigid body below. For the case of the rigid body, these equations read as follows. First, Equation (9.6.1) reads ∂SL ∂t +HL(−∇SL(A) ·AT ) = 0, (9.6.4) that is, ∂SL ∂t +HL ( − ∂SL ∂Aij Akj ) (9.6.5) (sum over j) where the action function SL is a function of an orthogonal matrix A and where we have identified tangent and cotangent spaces using the bi-invariant metric on the rotation group. This metric corresponds to the standard Euclidean metric on the Lie algebra, when identified with Euclidean 3-space. This identification maps the Lie algebra bracket to the cross product. The expression∇SL(A)·AT is a skew symmetric matrix, that is, it lies in the Lie algebra so(3), so it makes sense for HL to be evaluated on it. As usual, one has to be careful how the gradient (derivative) ∇SL is computed, since there is a constraint AAT = I involved. If it is computed naively in the coordinates of the ambient space of 3× 3 matrices, then one interprets the expression ∇SL(A) · AT using naive partial derivatives and skew symmetrizing the result; this projects the gradient to the constraint space, so produces the gradient of the constrained function. Equation (9.6.5) thus is the Hamilton–Jacobi equation for the dynam- ics of a rigid body written directly in body representation. The flow of the Hamiltonian is generated by SL in the following way: It is the transforma- tion of initial conditions at time t = 0 to a general t determined by first solving the equation Πˆ0 = −AT · ∇SL(A) (9.6.6) for the matrix A and then setting Π = AΠ0, where Πˆ = [Πij ] is the skew matrix associated to the vector Π in the usual way: Πˆ · v = Π× v. (Again, the right hand side of (9.6.6) is to be skew symmetrized if the derivative was taken in the naive way with the constraint ignored.) We have written the result in terms of the body angular momentum vector Π; one can rewrite it in terms of the body angular velocity vector by using the relation Π = Iω, where I is the moment of inertia tensor. In coordinates, equation (9.6.6) reads as follows: (Π0)ki = −Aji ∂SL ∂Ajk . (9.6.7) 186 9. Mechanical Integrators Finally, we note that similar equations also apply for fluids and plasmas, since they are also Lie–Poisson systems (but with right reduction). Also, the methods here clearly will generalize to the situation for reduction of any cotangent bundle; this generality is needed for example, for the case of free boundary fluids—see Lewis, Marsden, Montgomery, and Ratiu [1986]. One of the ideas of current interest is to use mechanical integrators on vortex algorithms. For example, one can use it on point vortices in the plane (Pullin and Saffman [1991]) or on vortex dipoles in three space (cf. Rouhi and Wright [1988]) both of which live on finite dimensional coadjoint orbits for the Euler equations (Marsden and Weinstein [1983]). A general way to construct first order algorithms valid in the Lie–Poisson setting (as well as its analogues in the symplectic and Poisson context) is as follows. Let H : g∗ → R be a given Hamiltonian function and let S0 be a function that generates a Poisson transformation ϕ0 : g∗ → g∗ and let S∆t = S0 + ∆tH(L∗gdS0). (9.6.8) For small ∆t, (9.6.8) generates a Poisson transformation, say ϕ∆t : g∗ → g∗. Then we have: 9.6.2 Proposition. With the assumptions above, the algorithm Πk 7→ Πk+1 = ϕ−10 ◦ ϕ∆t(Πk) (9.6.9) is a Poisson difference scheme that is a first order difference scheme for the Hamiltonian system with Hamiltonian H. In particular, if one can generate the identity transformation with a function S0, then one can get a specific first order scheme. On G, one can introduce singularities in the time variable to do this, as we have already re- marked. Interestingly, for g semisimple, one can do this in a non-singular way on g∗. In fact, in this case, the function S0(g) = trace(Ad∗g) (9.6.10) generates the identity in a G-invariant neighborhood of the zero of g∗. One can also check this with a direct calculation using (9.6.2) and (9.6.3). The neighborhood condition is necessary since there may be some restrictions on Π0 required for the solvability of (9.6.2). For example, for the rigid body the condition is checked to be ‖Π0‖ < 1. This condition can be dealt with using a scaling argument. We note that when one solves (9.6.2) for g, it need not be the identity, and consistent with (9.6.3) we observe that the solution g lies in the coadjoint isotropy of the element Π0. More generally, we say that a Lie algebra is a regular quadratic Lie algebra if there is a symmetric, Ad-invariant, non-degenerate bilinear form on g. 9.7 Example: The Free Rigid Body 187 9.6.3 Proposition. There is a function S0 defined in some neighborhood of the identity element in G that generates the identity map of g∗ iff g is a regular quadratic Lie algebra. Proof. If S0 exists, define the bilinear form B : g× g→ R by B(ξ, η) = d ds ∣∣∣∣ s=0 d dt ∣∣∣∣ t=0 S0(exp(sξ) exp(tη)) for ξ, η ∈ g. One verifies B is Ad-invariant and is non-degenerate from the fact that S0 generates the identity. Conversely, given B, define S0 by S0(g) = B(log g, log g) (9.6.11) where log is a local inverse of the exponential map. Combining (9.6.8) and (9.6.10) we get the following proposition. 9.6.4 Proposition. The generating function S∆t(g) = trace(Ad∗g) + ∆tH(L ∗ gd trace(Ad ∗ g)). (9.6.12) defines, via (9.6.2) and (9.6.3), a Poisson map that is a first order Poisson integrator for the Hamiltonian H. We remark that this scheme will automatically preserve additional con- served quantities on g∗ that, for example, arise from invariance of the Hamiltonian under a subgroup of G acting on the right . This is the situa- tion for a rigid body with symmetry and fluid flow in a symmetric container (with left and right swapped), for instance. 9.7 Example: The Free Rigid Body For the case of the free rigid body, we let so(3), the Lie algebra of SO(3), be the space of skew symmetric 3×3 matrices. An isomorphism between so(3) and R is given by mapping the skew vector v to the matrix vˆ defined previ- ously. Using the Killing form 〈A,B〉 = 12 trace(ATB), which corresponds to the standard inner product on R3, that is, 〈vˆ, wˆ〉 = v ·w, we identify so(3) with so(3)∗. We write the Hamiltonian H : so(3) → R as H(vˆ) = 12v · Iv, where I is the moment of inertia tensor. Let Iˆ : so(3) → so(3) be de- fined by Iˆ(vˆ) = (Iv)ˆ. Thus, H(vˆ) = 14 〈vˆ, Iˆ(vˆ)〉. Equation (9.6.10) becomes S0(A) = trace(A) and so TL∗AdS0 = 1 2 (A − AT ). Therefore, (9.6.12) be- comes S∆t(A) = trace(A) + ∆tH ( 1 2 (A−AT ) ) (9.7.1) 188 9. Mechanical Integrators and so Proposition 9.6.2 gives the following specific Lie–Poisson algorithm for rigid body dynamics: It is the scheme Πk 7→ Πk+1 defined by Πˆk = 1 2 [ 1 4 {AIˆ(A−AT ) + Iˆ(A−AT )AT }∆t+ (A−AT ) ] , (9.7.2) Πˆk+1 = 1 2 [ 1 4 {Iˆ(A−AT )A+AT Iˆ(A−AT )}∆t+ (A−AT ) ] (9.7.3) where, as before, the first equation is to be solved for the rotation matrix A and the result substituted into the second. Letting AS = 12 [A−AT ] denote the skew part of the matrix A, we can rewrite the scheme as Πˆk = AS + (AIˆAS)S∆t, (9.7.4) Πˆk+1 = AS + (AT IˆAS)S∆t. (9.7.5) Of course, one can write A = exp(ξ) and solve for ξ and express the whole algorithm in terms of g and g∗ alone. This type of algorithm does not keep track (except implicitly) of the rigid body phase. One can imagine combining the ideas here with those in Chapter 6 to do that. We know from the general theory that this scheme will automatically be Poisson and will, in particular, preserve the coadjoint orbits, that is, the total angular momentum surfaces ‖Π‖2 = constant. Of course, using other choices of S0, it is possible to generate other algorithms for the rigid body, but the choice S0(A) = trace(A) is particularly simple. We point out the interesting feature that the function (9.6.10) for the case of ideal Euler fluid flow is the function that assigns to a fluid placement field ϕ (an element of the diffeomorphism group of the containing region) the trace of the linear operator ω 7→ ϕ∗ω, on vorticity fields ω, which measures the vortex distortion due to the nonrigidity of the flow. (See Marsden and Weinstein [1983] for further information.) 9.8 PDE Extensions There have been some interesting developments in symplectic-type inte- grators for PDE’s. One point of view is to regard a PDE as an ordinary differential equation on an infinite dimensional space and to apply sym- plectic (or energy-momentum) algorithms to this context. Another point of view is to develop the local description of a PDE in terms of jet bundles. This point of view has been a very rich one within the development of the calculus of variations and has led to the subject of multisymplectic geome- try. See, for example, Marsden and Shkoller [1998] and references therein for this approach. Within this point of view, the variational technique of constructing sym- plectic integrators has been interesting. This allows, for example, one to 9.8 PDE Extensions 189 take a truly spacetime point of view in the development of symplectic inte- grators. This development is given in Marsden, Patrick and Shkoller [1998] who also show that this approach leads to very effective algorithms for specific PDE’s, such as the sine-Gordon equation. 190 9. Mechanical Integrators Page 191 10 Hamiltonian Bifurcation In this chapter, we study some examples of bifurcations in the Hamilto- nian context. A lot of the ideas from the previous chapters come into this discussion, and links with new ones get established, such as connections with chaotic dynamics and solution spaces in relativistic field theories. Our discussion will be by no means complete; it will focus on certain results of personal interest and results that fit in with the rest of the chapters. Some additional information on bifurcation theory in the Hamiltonian context may be found in the references cited below and in Abraham and Marsden [1978], Arnold [1989], Meyer and Hall [1991] and the references therein. 10.1 Some Introductory Examples Bifurcation theory deals with the changes in the phase portrait structure of a given dynamical system as parameters are varied. One usually begins by focussing on the simplest features of the phase portrait, such as equi- librium points, relative equilibria, periodic orbits, relative periodic orbits, homoclinic orbits, etc., and studies how they change in number and sta- bility characteristics as the system parameters are changed. Often these changes lead to new structures, such as more equilibria, periodic orbits, tori, or chaotic solutions, and the way in which stability or instability is transferred to these new structures from the old ones is of interest. As we pointed out in Chapter 2, the symmetry (isotropy) group of a point in phase space determines how degenerate it is for the momentum 192 10. Hamiltonian Bifurcation map. Correspondingly, one expects (see Golubitsky, Stewart, and Schaeffer [1988]), that these symmetry groups will play a vital role in the bifurca- tion theory of relative equilibria and its connections with dynamic stability theory. The beginnings of this theory have started and as it evolves, it will be tightly tied with the normal form methods and with the topology of the level sets of H × J and their associated bifurcations as the level set values and other system parameters vary. To begin, one can look at the case in which the symmetry group of a point is discrete, so that the point is a regular value of the momentum map. Already this case is reasonably rich; we shall comment on the more general case in §10.5 below. In the case of discrete symmetry groups, the energy momentum method can be used to help put the linearized system into normal form and from this one can calculate the stability transitions, and the eigenvalue evolution. To begin with a simple example, consider a ball moving with no friction in a hoop constrained to rotate with angular velocity ω (Figure 10.1.1). z x y R g = acceleration due to gravity Figure 10.1.1. A ball in a rotating hoop. In a moment we shall show that as ω increases past √ g/R, the stable equilibrium at θ = 0 becomes unstable through a Hamiltonian pitchfork bifurcation (Figure 10.1.2). The symmetry of these phase portraits is a reflection of the original Z2 symmetry of the mechanical system. One can break this symmetry by, for example, putting the rotation axis slightly off- center, as we shall discuss below. Breaking this symmetry is an example of system symmetry breaking since it is the whole system that loses the 10.1 Some Introductory Examples 193 symmetry. In Figure 10.1.2 notice that the stable solution at the origin has Z2 symmetry, before the bifurcation, but that the stable solutions for larger ω do not. This is an example of solution symmetry breaking within a symmetric system. increasing Figure 10.1.2. Hamiltonian pitchfork bifurcation as ω passes criticality. This example will be compared with bifurcations of a planar liquid drop (with a free boundary held with a surface tension τ) following Lewis, Mars- den, and Ratiu [1987] and Lewis [1989]. In this example, a circular drop loses its circular symmetry to a drop with Z2×Z2 symmetry as the angular momentum of the drop is increased (although the stability analysis near the bifurcation is somewhat delicate). There are also interesting stability and bifurcation results in the dynamics of vortex patches, especially those of Wan in a series of papers starting with Wan and Pulvirente [1984]. The ball in the hoop is an example of a steady state bifurcation. This situation is already complicated, even when only discrete symmetry of the underlying system is considered. A similar bifurcation occurs in the dy- namics of rotating planar coupled rigid bodies, as was analyzed by Oh, Sreenath, Krishnaprasad, and Marsden [1989] and in the bifurcations of 194 10. Hamiltonian Bifurcation two coupled rigid bodies, as in Patrick [1989, 1990]. It also occurs in our double spherical pendulum, as we shall see. Another basic bifurcation in the Hamiltonian context is the one-to-one resonance, or the Hamiltonian Hopf bifurcation. In this case, two eigenval- ues of the system linearized at a given equilibrium (or relative equilibrium) come together on the imaginary axis—that is, two frequencies in the sys- tem become equal. This is the Hamiltonian analogue of the well known Poincare´–Hopf bifurcation for nonhamiltonian vector fields. A classical example of the Hamiltonian Hopf bifurcation is the fast slow transition in an upright spinning heavy Lagrange top. When the top makes the transition from a fast top to a slow top ω ↓ 2 √ Mg`I1 I3 an instability sets in of this sort. For a study of the possible bifurcations in the dynamics of a heavy top see Lewis, Ratiu, Simo, and Marsden [1992]. Behavior of this sort is sometimes called a gyroscopic instability , or a Krein collision (see Krein [1950]). Here more complex dynamic behav- ior ensues, including periodic and chaotic motions (see also Holmes and Marsden [1983] and references therein for how chaotic motion is related to a homoclinic orbit that grows out of the fast-slow transition). Next, we give a few more details for the ball moving in a rotating hoop. The particle is assumed to have mass m and be acted on by gravitational and frictional forces, as well as constraint forces that keep it on the hoop. The hoop itself is spun about a vertical axis with constant angular velocity ω, as in Figure 10.1.3. The position of the ball in space is specified by the angles θ and ϕ, as shown. We can take ϕ = ωt, so the position of the ball becomes determined by θ alone. Let the orthonormal frame along the coordinate directions eθ, eϕ, and er be as shown. The forces acting on the particle are: 1. Friction, proportional to the velocity of the ball relative to the hoop: −νRθ˙eθ, where ν ≥ 0 is a constant. 2. Gravity: −mgk. 3. Constraint forces in the directions er and eϕ to keep the ball in the hoop. The equations of motion are derived from Newton’s second law F = ma. To get them, we calculate the acceleration a. Relative to the xyz coordinate system, we have x = R sin θ cosϕ y = R sin θ sinϕ (10.1.1) z = −R cos θ. 10.1 Some Introductory Examples 195 x z y R Figure 10.1.3. Coordinates for the derivation of the equations. Calculating the second derivatives using ϕ = ωt and the chain rule in (10.1.1) gives x¨ = −ω2x− θ˙2x+ (R cos θ cosϕ)θ¨ − 2Rωθ˙ cos θ sinϕ, y¨ = −ω2y − θ˙2y + (R cos θ sinϕ)θ¨ + 2Rωθ˙ cos θ cosϕ, (10.1.2) z¨ = −zθ˙2 +R sin θθ¨. If i,J,k denote unit vectors along the x, y, and z axes respectively, then eθ = (cos θ cosϕ)i + (cos θ sinϕ)J + sin θk. (10.1.3) In F = ma, F is the sum of the three forces described earlier and a = x¨i + y¨J + z¨k. (10.1.4) The eϕ and er components of F = ma tell us what the constraint forces must be; the equation of motion comes from the eθ component: F · eθ = ma · eθ. (10.1.5) Using (10.1.3), the left side of (10.1.5) is F · eθ = −νRθ˙ −mg sin θ (10.1.6) while from (10.1.1)–(10.1.4), the right side of (10.1.5) is, after some algebra, ma · eθ = mR{θ¨ − ω2 sin θcosθ}. (10.1.7) 196 10. Hamiltonian Bifurcation Comparing (10.1.5), (10.1.6), and (10.1.7), we get θ¨ = ω2 sin θ cos θ − ν m θ˙ − g R sin θ (10.1.8) as the equation of motion. Several remarks concerning (10.1.8) are in order: (i) If ω = 0 and ν = 0, (10.1.8) reduces to the pendulum equation Rθ¨ + g sin θ = 0. (10.1.9) In fact our system can be viewed just as well as a whirling planar pendulum . We notice that (10.1.8), when expressed in terms of the angular mo- mentum is formally the same as the reduction of the spherical pendu- lum. See equation (3.5.9). It is interesting that (10.1.8) is nonsingular near θ = 0, whereas (3.5.9) has a singularity. (ii) For ν = 0, (10.1.8) is Hamiltonian with respect to q = θ, p = mR2θ˙, the canonical bracket structure {F,K} = ∂F ∂q ∂K ∂p − ∂K ∂q ∂F ∂p and energy H = p2 2m −mgR cos θ − mR 2ω2 2 sin2 θ. (10.1.10) We can also use Lagrangian methods to derive (10.1.8). From the figure, the velocity is v = Rθ˙eθ + (ωR sin θ)eϕ, so T = 1 2 m‖v‖2 = 1 2 m(R2θ˙2 + [ωR sin θ]2) (10.1.11) while the potential energy is, V = −mgR cos θ, (10.1.12) so we choose L = T − V = 1 2 mR2θ˙2 + mR2ω2 2 sin2 θ +mgR cos θ. (10.1.13) The Euler–Lagrange equations d dt ∂L ∂θ˙ − ∂L ∂θ = T then give (10.1.8). The Legendre transform gives p = mR2θ˙ and the Hamil- tonian (10.1.10). 10.1 Some Introductory Examples 197 Consider equilibrium solutions; that is, solutions satisfying θ˙ = 0, and θ¨ = 0; (10.1.8) gives Rω2 sin θ cos θ = g sin θ. (10.1.14) Certainly θ = 0 and θ = pi solve (10.1.14) corresponding to the particle at the bottom or top of the hoop. If θ 6= 0 or pi, (10.1.14) becomes Rω2 cos θ = g (10.1.15) which has two solutions when g/(Rω2) < 1. The value ωc = √ g R (10.1.16) is the critical rotation rate . (Notice that ωc is the frequency of linearized oscillations for the simple pendulum that is, for Rθ¨ + gθ = 0.) For ω < ωc there are only two solutions θ = 0, pi, while for ω > ωc there are four solutions, θ = 0, pi,± cos−1 ( g Rω2 ) . (10.1.17) We say that a Hamiltonian pitchfork bifurcation occurs as ω crosses ωc. This system with ν = 0 is symmetric in the sense that the symplectic Z2- action given by θ 7→ −θ, and θ˙ 7→ −θ˙ leaves the phase portrait invariant. If this Z2 symmetry is broken, by setting the rotation axis a little off center, for example, then one side gets preferred, as in Figure 10.1.4. Let denote the off-center distance, so is a symmetry breaking parameter . Figure 10.1.4. The ball in the off-center hoop. The evolution of the phase portrait for 6= 0 is shown in Figure 10.1.5. Near θ = 0, the potential function has changed from the symmetric bifurcation in Figure 10.1.2 to the unsymmetric one in Figure 10.1.6. This is the cusp catastrophe . 198 10. Hamiltonian Bifurcation Figure 10.1.5. Phase portrait for the ball in the off-center hoop. (a) (b) Figure 10.1.6. Potential for the centered and off-center ball in the hoop as the angular velocity increases. Aside For the symmetric ball in the hoop, imagine that the hoop is subject to small periodic pulses; say ω = ω0 + cos(ηt), or perhaps the hoop is coupled to another oscillator. Using the Melnikov method described below, it is reasonable to expect that the resulting time periodic system has horseshoe chaos if and ν are both small, but /ν is large enough. 10.2 The Role of Symmetry Consider a Hamiltonian vector field XH on a phase space P depending on a parameter λ (like the angular velocity of the hoop in Figure 10.1.1) and we have a given curve z(λ) of equilibrium solutions—these can also be relative equilibria if we work on the reduced phase space. If we linearize the vector field at the equilibrium we get a linear Hamiltonian system on the tangent space Tz(λ)P and we can examine its eigenvalues. For relative equilibria that have at most discrete symmetry, one can apply the same procedure to the reduced vector field, where one sees a genuine equilibrium. The possible movement of eigenvalues that we focus on are illustrated by the following two cases: 10.2 The Role of Symmetry 199 1. (Steady-state bifurcation) The equilibrium has a zero eigenvalue of multiplicity two. 2. (One-to-one resonance) The equilibrium has a pair of purely imagi- nary eigenvalues of multiplicity 2. (Without loss of generality, we may assume that these eigenvalues are ±i.) In Case 1, the kernel of the linearization is a two-dimensional symplectic subspace. As the bifurcation parameter is varied, generically the eigenvalues go from purely imaginary to real, or vice-versa. In Case 2, the sum of the eigenspaces of the eigenvalues ±i can be written as the sum of two ω-orthogonal two-dimensional symplectic subspaces. This time, generically the eigenvalues go from purely imaginary into the right and left hand complex plane, or vice versa. In each of these cases we say that the eigenvalues split , (see Figure 10.2.1). The one-to-one resonance with splitting is often called the Hamiltonian Hopf bifurcation (see van der Meer [1985]). (a) (b) i -i i -i i -i Figure 10.2.1. The splitting case; (a) for the steady state bifurcation, (b) for the one-to-one resonance. In some applications the eigenvalues do not split at 0 or ±i, but rather, they remain on the imaginary axis and pass, as in Figure 10.2.2. Symmetry can influence the above generic behavior (see Golubitsky, Stewart, and Schaeffer [1988]). Indeed, for certain symmetry groups (such as the circle group S1), the passing of eigenvalues may be generic in a one parameter family. Galin [1982] shows that without symmetry, the generic situation is splitting and one would require three parameters to see passing. 200 10. Hamiltonian Bifurcation (a) (b) i -i i -i i -i Figure 10.2.2. The passing case; (a) for the steady state bifurcation, (b) for the one-to-one resonance. In the steady state case, the dichotomy in eigenvalue movements can be understood using definiteness properties of the Hamiltonian, that is, by energetics, or group theoretically . The group theoretic approach was discussed in Golubitsky and Stewart [1987]. The energetics method has a complex history, going back to at least Cartan [1922]—a modern reference is Oh [1987]. For example, one can sometimes say that a S1 symmetry forces the eigenvalues to stay on the imaginary axis, or one can say that eigenvalues must split because the second variation changes from positive definite to indefinite with just one eigenvalue crossing through zero. These methods have also been used extensively in the study of solitons; see Pego and Weinstein [1992] and references therein. Here are some simple observations about the method of energetics. Start with a linear (or linearized) Hamiltonian system and write it in the form XH = JB where B is a symmetric matrix whose associated quadratic form is the Hamiltonian H and where J = [ 0 1−1 0 ] is the Poisson tensor. If H is positive (or negative) definite, then the spectrum of XH is on the imaginary axis—this follows because the system is necessarily stable, and the spec- trum of XH is symmetric under reflection in the two axis of the complex plane, so the spectrum must be confined to the imaginary axis. If H has an odd number of eigenvalues that are negative, then taking the determinant, we see that since J has determinant one, XH has a negative determinant. Thus in this case, it must have at least one pair of real eigenvalues, and therefore be linearly unstable. If H has the standard form of kinetic plus potential energy, and the kinetic term is positive definite, and the potential 10.2 The Role of Symmetry 201 energy has at least one negative eigenvalue, then again XH has real eigen- values, and so is linearly unstable. However, for gyroscopic systems, such as those that arise in reduction, the situation is not so simple, and deeper insight is needed. An example relevant to the above remarks concerns bifurcations of rel- ative equilibria of a rotating liquid drop: the system consists of the two dimensional Euler equations for an ideal fluid with a free boundary. A rigidly rotating circular drop is an equilibrium solution (in the spatially reduced equations). The energy–Casimir method shows stability, provided Ω2 12R3τ < 1. (10.2.1) Here Ω is the angular velocity of the circular drop, R is its radius and τ is the surface tension, a constant. As Ω increases and (10.2.1) is violated, the stability of the circular solution is lost and is picked up by elliptical-like solutions with Z2 × Z2 symmetry. (The bifurcation is actually subcritical relative to Ω and is supercritical relative to the angular momentum.) This is proved in Lewis, Marsden, and Ratiu [1987] and Lewis [1989], where other references may also be found (see Figure 10.2.3). circular stable solutions uniform ly rotating elliptical-like solutions increasing angular m om entum R Figure 10.2.3. The bifurcation of the rotating planar liquid drop. During this transition, the eigenvalues stay on the imaginary axis—they are forced to because of the symmetry. This is consistent with the energetics approach since an even number of eigenvalues of the second variation cross through zero, namely two. The situation for the ball in the hoop and the liquid drop examples is presented in Figure 10.2.4. Energetics or group theory alone is not sufficient to characterize the movement of eigenvalues in the one-to-one resonance. The work of Dellnitz, Melbourne, and Marsden [1992] uses a combination of group theory and energetics that gives a particularly clean characterization of the splitting and passing cases. We summarize some relevant notation to explain these results. Assume that the quadratic Hamiltonian is invariant under a compact Lie group Γ that preserves the symplectic structure. A Γ-invariant subspace V 202 10. Hamiltonian Bifurcation y x y x (a) with sym m etry (b) without sym m etry Figure 10.2.4. Eigenvalue evolution for the liquid drop and the ball in the hoop. is called absolutely Γ-irreducible if the only linear mappings V → V that commute with the action of Γ are real multiples of the identity. An Γ-irreducible subspace that is not absolutely Γ-irreducible is called non- absolutely Γ-irreducible . For example, the rotation group SO(2) acting on the plane is nonabsolutely irreducible since any rotation commutes with this action, but nevertheless, the action is irreducible (has no nontrivial invariant subspaces). On the other hand, the rotation group SO(3) acting in the usual way on three space is absolutely Γ-irreducible, as is easy to check. Golubitsky and Stewart [1987] show that for steady-state bifurcation, generically the generalized zero eigenspace E0 is either nonabsolutely Γ- irreducible or the direct sum of two isomorphic absolutely Γ-irreducible subspaces. These two possibilities correspond respectively to the passing or splitting of eigenvalues. In terms of energetics, the Hamiltonian changes from definite to indefinite in the splitting case, but remains definite in the passing case. In the case of one-to-one resonance, generically the sum of the general- ized eigenspaces of ±i, E±i, can be written as the sum of two symplectic ω-orthogonal subspaces U1 and U2, where each of the Uj is either non- absolutely Γ-irreducible or the direct sum of two isomorphic absolutely Γ-irreducible subspaces. The main result of Dellnitz, Melbourne, and Marsden [1992] concerns the generic movement of eigenvalues in this situation. The most difficult cases are when U1 and U2 are isomorphic. In fact, if U1 and U2 carry distinct representations of Γ then the reso- nance decouples and the eigenvalues move independently along the imaginary axis (independent passing). 10.3 The One-to-One Resonance and Dual Pairs 203 To understand the cases where U1 and U2 are isomorphic, one uses the results of Montaldi, Roberts, and Stewart [1988,1990] on the relationship between the symmetry and the symplectic structure. At this stage it be- comes necessary to distinguish between the two types of nonabsolutely Γ-irreducible representations: complex and quaternionic. Here, one uses the fact that for a Γ-irreducible representation, the space of linear map- pings that commute with Γ is isomorphic to the reals, complexes, or to the quaternions. The real case corresponds to the absolutely irreducible case, and in the nonabsolutely irreducible case, one has either the complexes or the quaternions. If the Uj are isomorphic complex irreducibles, then in the terminology of Montaldi, Roberts, and Stewart [1988], they are either of the same type or dual . When they are of the same type, one has a sym- plectic form that can be written as J = [ i 00 i ] and in the case of duals, the symplectic form can be written as J = [ i 0 0 −i ] . There are three cases: 1. Provided U1 and U2 are not complex irreducibles, generically the eigenvalues split and H is indefinite. 2. If U1 and U2 are complex of the same type, then generically the eigenvalues pass and H is indefinite. 3. In the case of complex duals the eigenvalues can generically pass or split and these possibilities correspond precisely to definiteness and indefiniteness of the quadratic form induced on U1 ⊕ U2 by the linearization. 10.3 The One-to-One Resonance and Dual Pairs An interesting discussion of the one-to-one resonance is given in Cushman and Rod [1982] and Marsden [1987]. Consider a family of Hamiltonians depending on a parameter λ near one of the form H = 1 2 (q21 + p 2 1) + λ 2 (q22 + p 2 2) + higher order terms. (10.3.1) Notice that the quadratic part of this Hamiltonian has a S1×S1 symmetry acting separately on both factors and so we have, in the terminology of the preceding section, independent passing. Notice that H is definite in this case. The oscillators have the same frequency when λ = 1, corresponding to the one-to-one resonance. To analyze the dynamics of H, it is important to utilize a good geometric picture for the critical case when λ = 1 and we get the unperturbed Hamiltonian H0 = 1 2 (q21 + p 2 1 + q 2 2 + p 2 2). (10.3.2) 204 10. Hamiltonian Bifurcation The energy level H0 = constant is the three sphere S3 ⊂ R4. If we think of H0 as a function on C2 by letting z1 = q1 + ip1 and z2 = q2 + ip2, then H0 = (|z1|2+|z2|2)/2 and so H0 is invariant under the action of SU(2), the complex 2× 2 unitary matrices of determinant one. The corresponding conserved quantities are W1 = 2(q1q2 + p1p2) W2 = 2(q2p1 − q1p2) (10.3.3) W3 = q21 + p 2 1 − q22 − p22 which comprise the components of a (momentum) map for the action of SU(2) on C2: J : C2 ∼= R4 → su(2)∗ ∼= R3. (10.3.4) From the relation 4H20 = W 2 1 +W 2 2 +W 2 3 , one finds that J restricted to S 3 gives a map j : S3 → S2. (10.3.5) The fibers j−1(point) are circles and the dynamics of H0 moves along these circles. The map j is the Hopf fibration which describes S3 as a topologi- cally nontrivial circle bundle over S2. (The reduction of R4 by the action by the flow of H0 is S2.) Apparently the role of the Hopf fibration in mechanics was already known to Reeb around 1950. With P = C2, we have a basic example of a dual pair determined by the above momentum maps (see Example A of §8.3 and the Example in §8.5). See Figure 10.3.1. C2 J H0 su(2)∗ R @ @ @ @R Figure 10.3.1. The dual pair of the harmonic oscillator. Normal form theory allows one (up to finite order) to change coordinates by averaging over the S1 action determined by the flow of H0. In this way one gets a new Hamiltonian H from H that is S1 invariant. Since we have a dual pair, such an H can be written as H = h ◦ J. (10.3.6) In other words, a function invariant on one side collectivizes on the other . In particular, since J is a Poisson map, the dynamics of H can be reduced 10.4 Bifurcations in the Double Spherical Pendulum 205 to dynamics on su(2)∗ ∼= R3 with the rigid body Lie–Poisson structure. This proves one of the results of Cushman and Rod [1982]. This procedure can be of help in locating interesting bifurcations, as in David, Holm, and Tratnik [1990]. The Hopf fibration occurs in a number of other interesting mechanical systems. One of these is the free rigid body. When doing reduction for the rigid body, we construct the reduced space J−1(µ)/Gµ = J−1(µ)/S1, which is the sphere S2. Also, J−1(µ) is topologically the same as the rotation group SO(3), which in turn is the same as S3/Z2. Thus, the reduction map is a map of SO(3) to S2. Such a map is given explicitly by taking an orthogonal matrix A and mapping it to the vector on the sphere given by Ak, where k is the unit vector along the z-axis. This map that does the projection is in fact a restriction of a momentum map and, when composed with the map of S3 ∼= SU(2) to SO(3), is just the Hopf fibration again. Thus, not only does the Hopf fibration occur in the one-to-one resonance, it occurs in the rigid body in a natural way as the reduction map from material to body representation! 10.4 Bifurcations in the Double Spherical Pendulum In §5.5 we wrote the equations for the linearized solutions of the double spherical pendulum at a relative equilibrium in the form Mq¨ + Sq˙ + Λq = 0 (10.4.1) for certain 3×3 matricesM,S and Λ. These equations have the Hamiltonian form F˙ = {F,H} where p = Mq˙, H = 1 2 pM−1P + 1 2 qΛq (10.4.2) and {F,K} = ∂F ∂qi ∂K ∂pi − ∂K ∂qi ∂F ∂pi − Sij ∂F ∂pi ∂K ∂pj ; (10.4.3) that is, q˙ = M−1p p˙ = −Sq˙ − Λq = −SM−1p− Λq. (10.4.4) The following is a standard useful observation: 10.4.1 Proposition. The eigenvalues λ of the linear system (10.4.4) are given by the roots of det[λ2M + λS + Λ] = 0. (10.4.5) 206 10. Hamiltonian Bifurcation Proof. Let (u, v) be an eigenvector of (10.4.4) with eigenvalue λ; then M−1v = λu and − SM−1v − Λu = λv that is, −Sλu−Λu = λ2Mu, so u is an eigenvector of λ2M + λS + Λ. For the double spherical pendulum, we call the eigenvalue γ (since λ is already used for something else in this example) and note that the polyno- mial p(γ) = det[γ2M + γS + Λ] (10.4.6) is cubic in γ2, as it must be, consistent with the symmetry of the spec- trum of Hamiltonian systems. This polynomial can be analyzed for specific system parameter values. In particular, for r = 1 and m = 2, one finds a Hamiltonian Hopf bifurcation along the cowboy branch as we go up the branch in Figure 4.3.1 with increasing λ starting at α = −√2. The situation before the bifurcation (for smaller λ and µ), is one where the energetics method and the spectral method disagree in their conclusions about stability. The situation will be resolved in §10.7. Perhaps more interesting is the fact that for certain system parameters, the Hamiltonian Hopf point can converge to the straight down singular(!) state with λ = 0 = µ. Here, (10.4.1) does not make sense, and must be regu- larized . After this is done, one finds (with still two parameters left) that one has a system in which both passing and splitting can generically occur. One can hope that the ideas of §10.2 with the inclusion of an antisymplectic re- versibility type of symmetry will help to explain this observed phenomenon. We refer to Dellnitz, Marsden, Melbourne, and Scheurle [1992] for further details. 10.5 Continuous Symmetry Groups and Solution Space Singularities Recall that singular points of J are points with symmetry . This turns out to be a profound observation with far reaching implications. The level sets of J typically have quadratic singularities at its singular (= symmetric) points, as was shown by Arms, Marsden, and Moncrief [1981]. In the Abelian case, the images of these symmetric points are the vertices, edges and faces of the convex polyhedron J(P ) in the Atiyah–Guillemin–Sternberg convexity theory. (See Atiyah [1982] and Guillemin and Sternberg [1984].) As one leaves this singular point, heading for a generic one with no singularities, one passes through the lattice of isotropy subgroups of G. Arms, Marsden, and Moncrief [1981] describe how these symmetry groups break. These ideas apply in a remarkable way to solution spaces of relativis- tic field theories, such as Einstein’s equations of general relativity and 10.6 The Poincare´–Melnikov Method 207 the Yang–Mills equations on space time. Here the theories have symme- try groups and, appropriately interpreted, corresponding momentum maps. The relativistic field equations split into two parts—Hamiltonian hyper- bolic evolution equations and elliptic constraint equations. The solution space structure is determined by the elliptic constraint equations, which in turn say exactly that the momentum map vanishes. A fairly long story of both geometry and analysis is needed to establish this, but the result can be simply stated: the solution space has a quadratic singularity precisely at those field points that have symmetry . For further details, see Fischer, Marsden, and Moncrief [1980] and Arms, Marsden, and Moncrief [1982]. While these results were motivated by perturbation theory of classical solutions (gravitational waves as solutions of the linearized Einstein equa- tions etc.), there is some evidence that these singularities have quantum implications. For example, there appears to be evidence that in the Yang– Mills case, wave functions tend to concentrate near singular points (see, for example, Emmrich and Ro¨mer [1990]). For bifurcation theory, as we have indicated in the preceding sections, a start has been made on how to tackle the problem when there is a continu- ous isotropy group and some examples have been worked out. One of these is the bifurcations in a rotating liquid drop, already mentioned above, where the isotropy group is the whole symmetry group S1. Here the problems with the singular structure of the momentum map are obviated by working with the spatially reduced system, and the energy–Casimir method. Here, the symmetry is dealt with by directly factoring it out “by hand”, using ap- propriately defined polar coordinates. Another case that is dealt with is the heavy top in Lewis, Ratiu, Simo, and Marsden [1992]. Here, bifurca- tions emanate from the upright position, which has a nondiscrete symmetry group S1. That paper and Lewis [1992] indicate how a general theory of stability might go in the presence of general isotropy groups. Presumably the general bifurcation theory will follow. 10.6 The Poincare´–Melnikov Method To begin with a simple example, consider the equation of a forced pendulum φ¨+ sinφ = cosωt. (10.6.1) Here ω is a constant angular forcing frequency, and is a small parameter. For = 0 this has the phase portrait of a simple pendulum. For small but non-zero, (10.6.1) possesses no analytic integrals of the motion. In fact, it possesses transversal intersecting stable and unstable manifolds (separa- trices); that is, the Poincare´ maps Pt0 : R2 → R2 that advance solutions by one period T = 2pi/ω starting at time t0 possess transversal homoclinic 208 10. Hamiltonian Bifurcation points. This type of dynamic behavior has several consequences, besides precluding the existence of analytic integrals, that lead one to use the term “chaotic”. For example, Equation (10.6.1) has infinitely many periodic so- lutions of arbitrarily high period. Using the shadowing lemma, one sees that given any bi-infinite sequence of zeros and ones (for example, use the binary expansion of e or pi), there exists a corresponding solution of (10.6.1) that successively crosses the plane φ = 0 (the pendulum’s vertically downward configuration) with φ > 0 corresponding to a zero and φ < 0 corresponding to a one. The origin of this chaos on an intuitive level lies in the motion of the pendulum near its unperturbed homoclinic orbit—the orbit that does one revolution in infinite time. Near the top of its motion (where φ = ±pi) small nudges from the forcing term can cause the pendulum to fall to the left or right in a temporally complex way. The Poincare´–Melnikov method is as follows: First, write the dynamical equation to be studied in abstract form as x˙ = X0(x) + X1(x, t) (10.6.2) where x ∈ R2, X0 is a Hamiltonian vector field with energy H0, X1 is Hamiltonian with energy a T -periodic function H1. Assume that X0 has a homoclinic orbit x¯(t) so x¯(t)→ x0, a hyperbolic saddle point, as t→ ±∞. Second, compute the Poincare´–Melnikov function defined by M(t0) = ∫ ∞ −∞ {H0, H1}(x¯(t− t0), t)dt (10.6.3) where { , } denotes the Poisson bracket. 10.6.1 Theorem (Poincare´–Melnikov). If M(t0) has simple zeros as a function of t0, then (10.6.2) has, for sufficiently small , homoclinic chaos in the sense of transversal intersecting separatrices. We shall give a proof of this result below and in the course of the proof, we shall clarify what it means to have transverse separatrices. To apply this method to equation (10.6.1), let x = (φ, φ˙) so (10.6.1) becomes d dt [ φ φ˙ ] = [ φ˙ − sinφ ] + [ 0 cosωt ] . (10.6.4) The homoclinic orbits for = 0 are computed to be given by x¯(t) = [ φ(t) φ˙(t) ] = [ ±2 tan−1(sinh t) ±2 sech t ] and one has H0(φ, φ˙) = 1 2 φ˙2 − cosφ and H1(φ, φ˙, t) = φ cosωt. 10.6 The Poincare´–Melnikov Method 209 Hence (10.6.3) gives M(t0) = ∫ ∞ −∞ ( ∂H0 ∂φ ∂H1 ∂φ˙ − ∂H0 ∂φ˙ ∂H1 ∂φ ) (x¯(t− t0), t)dt = − ∫ ∞ −∞ φ˙(t− t0) cosωtdt = ∓ ∫ ∞ −∞ [2 sech (t− t0) cosωt]dt. (10.6.5) Changing variables and using the fact that sech is even and sin is odd, we get M(t0) = ∓2 (∫ ∞ −∞ sech t cosωtdt ) cos(ωt0). The integral is evaluated by residues: M(t0) = ∓2pi sech (piω 2 ) cos(ωt0), which clearly has simple zeros. Thus, this equation has chaos for small enough. Now we turn to a proof of the Poincare´–Melnikov theorem. Proof. There are two convenient ways of visualizing the dynamics of (10.6.2). Introduce the Poincare´ map P s : R2 → R2, which is the time T map for (10.6.2) starting at time s. For = 0, the point x0 and the homoclinic orbit are invariant under P s0 , which is independent of s. The hyperbolic saddle x0 persists as a nearby family of saddles x for > 0, small, and we are interested in whether or not the stable and unstable manifolds of the point x for the map P s intersect transversally (if this holds for one s, it holds for all s). If so, we say (10.6.2) admits horseshoes for > 0. The second way to study (10.6.2) is to look directly at the suspended system on R2 × S1: x˙ = X0(x) + X1(x, θ), (10.6.6) θ˙ = 1. (10.6.7) From this point of view, the curve γ0(t) = (x0, t) is a periodic orbit for (10.6.2), whose stable and unstable manifolds W s0 (γ0) and Wu0 (γ0) are coincident. For > 0 the hyperbolic closed orbit γ0 per- turbs to a nearby hyperbolic closed orbit which has stable and unstable manifolds W s (γ) and W u (γ). If W s (γ) and W u (γ) intersect transver- sally, we again say that (10.6.2) admits horseshoes. These two definitions of admitting horseshoes are equivalent. 210 10. Hamiltonian Bifurcation We use the energy function H0 to measure the first order movement of W s (γ) at x¯(0) at time t0 as is varied. Note that points of x¯(t) are regular points for H0 since H0 is constant on x¯(t) and x¯(0) is not a fixed point. Thus, the values of H0 can be used to measure the distance from the homoclinic orbit. If (xs(t, t0), t) is the curve on W s (γ) that is an integral curve of the suspended system in xt-space, and has an initial condition xs(t0, t0) which is the perturbation of W s0 (γ0) ∩ { the plane t = t0} in the normal direction to the homoclinic orbit, then H0(xs(t0, t0)) measures the normal distance. But H0(xs(T, t0))−H0(xs(t0, t0)) = ∫ T t0 d dt H0(xs(t, t0))dt, (10.6.8) and so H0(xs(T, t0))−H0(xs(t0, t0)) = ∫ T t0 {H0, H0+H1}(xs(t, t0), t)dt. (10.6.9) Since xs(T, t0) is -close to x¯(t− t0) (uniformly as T → +∞), d(H0 + H1)(xs(t, t0), t)→ 0 exponentially as t→ +∞, and {H0, H0} = 0, so (10.6.9) becomes H0(xs(T, t0))−H0(xs(t0, t0)) = ∫ T t0 {H0, H1}(x¯(t− t0, t))dt+O(2). (10.6.10) Similarly, H0(xu (t0, t0))−H0(xu (−S, t0)) = ∫ t0 −S {H0, H1}(x¯(t− t0, t))dt+O(2). (10.6.11) Since xs(T, t0)→ γ, a periodic orbit for the perturbed system as T → +∞, we can choose T and S such that H0(xs(T, t0)) − H0(xu (−S, t0)) → 0 as T, S →∞. Thus, adding (10.6.10) and (10.6.11), and letting T, S →∞, we get H0(xu (t0, t0))−H0(xs(t0, t0)) = ∫ ∞ −∞ {H0, H1}(x¯(t− t0, t))dt+O(2). (10.6.12) It follows that if M(t0) has a simple zero at time t0, then xu (t0, t0) and xs(t0, t0) intersect transversally near the point x¯(0) at time t0. (Since dH0 → 0 exponentially at the saddle points, the integrals involved in this criterion are automatically convergent.) We now describe a few of the extensions and applications of this tech- nique. The literature in this area is growing very quickly and we make no 10.6 The Poincare´–Melnikov Method 211 claim to be comprehensive (the reader can track down many additional references by consulting Wiggins [1988] and the references cited below). If in (10.6.2), X0 only is a Hamiltonian vector field, the same conclusion holds if (10.6.3) is replaced by M(t0) = ∫ ∞ −∞ (X0 ×X1)(x¯(t− t0), t)dt, (10.6.13) where X0×X1 is the (scalar) cross product for planar vector fields. In fact, X0 need not even be Hamiltonian if an area expansion factor is inserted. For example, (10.6.13) applies to the forced damped Duffing equation u¨− βu+ αu3 = (γ cosωt− xu˙). (10.6.14) Here the homoclinic orbits are given by u¨(t) = ± √ β α sech (β 1 2 t) (10.6.15) and (10.6.13) becomes, after a residue calculation, M(t0) = 2γpiω √ 2 α sech ( piω 2 √ β ) sin(ωt0) + 4δβ 3 2 3α (10.6.16) so one has simple zeros and hence chaos of the horseshoe type if γ δ > √ 2β 3 2 3ω √ α cosh ( piω 2 √ β ) (10.6.17) and is small. Another interesting example, due to Montgomery, concerns the equations for superfluid 3He. These are the Leggett equations and we shall confine ourselves to the A phase for simplicity. The equations are s˙ = −1 2 ( χΩ2 γ2 ) sin 2θ and θ˙ = ( γ2 χ ) s− (γB sinωt+ 1 2 Γ sin 2θ). (10.6.18) Here s is the spin, θ an angle (describing the order parameter) and γ, χ, . . . are physical constants. The homoclinic orbits for = 0 are given by θ¯± = 2 tan−1(e±Ωt)− pi/2 and s¯± = ±2 Ωe ±2Ωt 1 + e±2Ωt . (10.6.19) One calculates, after substituting (10.6.19) and (10.6.20) in (10.6.13) that M±(t0) = ∓piχωB8γ sech (ωpi 2Ω ) cosωt− 2 3 χ γ2 ΩΓ (10.6.20) 212 10. Hamiltonian Bifurcation so that (10.6.18) has chaos in the sense of horseshoes if γB Γ > 16 3pi Ω ω cosh (piω 2Ω ) (10.6.21) and if is small. A version of the Poincare´–Melnikov theorem applicable to PDE’s (due to Holmes and Marsden [1981]). One basically still uses formula (10.6.13) where X0 ×X1 is replaced by the symplectic pairing between X0 and X1. However, there are two new difficulties in addition to standard technical analytic problems that arise with PDEs. The first is that there is a se- rious problem with resonances. These can be dealt with using the aid of damping—the undamped case would need an infinite dimensional version of Arnold diffusion. Secondly, the problem is not reducible to two dimen- sions; the horseshoe involves all the modes. Indeed, the higher modes do seem to be involved in the physical buckling processes for the beam model discussed next. A PDE model for a buckled forced beam is w¨ + w′′′′ + Γw′ −K (∫ 1 0 [w′]2dz ) w′′ = (f cosωt− δw˙) (10.6.22) where w(z, t), 0 ≤ z ≤ 1 describes the deflection of the beam, ˙ = ∂/∂t, ′ = ∂/∂z and Γ,K, . . . are physical constants. For this case, the theory shows that if 1. pi2 < Γ < 4ρ3 (first mode is buckled), 2. j2pi2(j2pi2 − Γ) 6= ω2, j = 2, 3, . . . (resonance condition), 3. f δ > pi(Γ− pi2) 2ω √ K cosh ( ω 2 √ Γ− ω2 ) (transversal zeros for M(t0)), 4. δ > 0, and is small, then (10.6.22) has horseshoes. Experiments (see Moon and Holmes [1979]) showing chaos in a forced buckled beam provided the mo- tivation which lead to the study of (10.6.22). This kind of result can also be used for a study of chaos in a van der Waal fluid ( Slemrod and Marsden [1985]) and there is a growing literature using these methods for forced and damped soliton equations. For example, in the damped, forced Sine–Gordon equation one has chaotic transitions between breathers and kink-antikink pairs and in the Benjamin–Ono equation one can have chaotic transitions between solutions with different numbers of poles. For Hamiltonian systems with two degrees of freedom, Holmes and Mars- den [1982a] show how the Melnikov method may be used to prove the ex- istence of horseshoes on energy surfaces in nearly integrable systems. The 10.6 The Poincare´–Melnikov Method 213 class of systems studied have a Hamiltonian of the form H(q, p, θ, I) = F (q, p) +G(I) + H1(q, p, θ, I) +O(2) (10.6.23) where (θ, I) are action-angle coordinates for the oscillator G; G(0) = 0, G′ > 0. It is assumed that F has a homoclinic orbit x¯(t) = (q¯(t), p¯(t)) and that M(t0) = ∫ ∞ −∞ {F,H1}dt, (10.6.24) the integral taken along (x¯(t− t0),Ωt, I), has simple zeros. Then (10.6.23) has horseshoes on energy surfaces near the surface corresponding to the homoclinic orbit and small I; the horseshoes are taken relative to a Poincare´ map strobed to the oscillator G. The paper Holmes and Marsden [1982a] also studies the effect of positive and negative damping. These results are related to those for forced one degree of freedom systems since one can often reduce a two degrees of freedom Hamiltonian system to a one degree of freedom forced system. For some systems in which the variables do not split as in (10.6.23), such as a nearly symmetric heavy top, one can exploit symmetry of the system and make use of reduction ideas. The general theory for this is given in Holmes and Marsden [1983] and was applied to show the existence of horseshoes in the nearly symmetric heavy top; see also some closely related results of Ziglin [1981]. The Poincare´–Melnikov theory has been used by Ziglin [1980] in vortex dynamics, for example to give a proof of the non-integrability of the re- stricted four vortex problem. There have also been recent applications to the dynamics of general relativity showing the existence of horseshoes in Bianchi IX models. See Oh, Sreenath, Krishnaprasad, and Marsden [1989] for applications to the dynamics of coupled planar rigid bodies and to David, Holm, and Tratnik [1990] to the study of polarization laser dynam- ics. Arnold [1964] extended the Poincare´–Melnikov theory to systems with several degrees of freedom. In this case the transverse homoclinic man- ifolds are based on KAM tori and allow the possibility of chaotic drift from one torus to another. This drift, now known as Arnold diffusion is a basic ingredient in the study of chaos in Hamiltonian systems (see for instance, Chirikov [1979] and Lichtenberg and Liebermann [1983] and references therein). Instead of a single Melnikov function, one now has a Melnikov vector given schematically by → M=  ∫∞ −∞{H0, H1}dt∫∞ −∞{Ik, H1}dt  (10.6.25) where Ik are integrals for the unperturbed (completely integrable) system and where → M depends on t0 and on angles conjugate to I1, . . . , In. One 214 10. Hamiltonian Bifurcation requires → M to have transversal zeros in the vector sense. This result was given by Arnold for forced systems and was extended to the autonomous case by Holmes and Marsden [1982b, 1983]; see also Robinson [1988]. These results apply to systems such as pendulum coupled to several oscillators and the many vortex problems. It has also been used in power systems by Salam, Marsden, and Varaiya [1983], building on the horseshoe case treated by Kopell and R. B. Washburn [1982]. There have been a number of other directions of research on these techniques. For example, Grundler devel- oped a multidimensional version applicable to the spherical pendulum and Greenspan and Holmes showed how it can be used to study subharmonic bifurcations. See Wiggins [1988] references and for more information. In Poincare´’s celebrated memoir (Poincare´ [1890]) on the three-body problem, he introduced the mechanism of transversal intersection of separa- trices which obstructs the integrability of the equations and the attendant convergence of series expansions for the solutions. This idea was subse- quently developed by Birkhoff and Smale using the horseshoe construction to describe the resulting chaotic dynamics. However, in the region of phase space studied by Poincare´, it has never been proved (except in some generic sense that is not easy to interpret in specific cases) that the equations really are nonintegrable. In fact Poincare´ himself traced the difficulty to the pres- ence of terms in the separatrix splitting which are exponentially small. A crucial component of the measure of the splitting is given by the following formula of Poincare´ [1890, p. 223]: J = −8pii exp ( pi√ 2µ ) + exp ( − pi√ 2µ ) which is exponentially small (or beyond all orders) in µ. Poincare´ was well aware of the difficulties that this exponentially small behavior causes; on p. 224 of his article, he states: “En d’autres termes, si on regarde µ comme un infiniment petit du premier ordre, la distance BB′ sans eˆtre nulle, est un infiniment petit d’ordre infini. C’est ainsi que la fonction e−1/µ est un infiniment petit d’ordre infini sans e`tre nulle . . . Dans l’example particulier que nous avons traite´ plus haut, la distance BB′ est du me`me ordre de grandeur que l’integral J , c’est a` dire que exp(−pi/√2µ).” This is a serious difficulty that arises when one uses the Melnikov method near an elliptic fixed point in a Hamiltonian system or in bifurcation prob- lems giving birth to homoclinic orbits. The difficulty is related to those described by Poincare´ (see Sanders [1982]). Near elliptic points, one sees homoclinic orbits in normal forms and after a temporal rescaling, this leads to a rapidly oscillatory perturbation that is modelled by the following vari- ation of (10.6.1): φ¨+ sinφ = cos ( ωt ) . (10.6.26) 10.6 The Poincare´–Melnikov Method 215 If one formally computes M(t0) one finds from (10.6.3): M(t0, ) = ±2pi sech (piω 2 ) cos ( ωt0 ) . (10.6.27) While this has simple zeros, the proof of the Poincare´–Melnikov theorem is no longer valid since M(t0, ) is now of order e−pi/2 and the error analysis in the proof only gives errors of order 2. In fact, no expansion in powers of can detect exponentially small terms like e−pi/2. Holmes, Marsden, and Scheurle [1988], Scheurle [1989], Scheurle, Mars- den, and Holmes [1991] and Delshams and Seara [1991] show that (10.6.26) has chaos that is, in appropriate sense exponentially small in . Not only that, examples show how truly subtle this situation is, and one has to be extremely careful with hidden assumptions in the literature (like a priori hypotheses about the order of magnitude of the splitting being of the form ke−c/ for some k and ) that can be false in some examples. To get such estimates, the extension of the system to complex time plays a crucial role. One can hope that if sharp enough such results for (10.6.26) can really be proven, then it may be possible to return to Poincare´’s 1890 work and complete the arguments he left unfinished. To illustrate how exponentially small phenomena enter bifurcation prob- lems, consider the problem of a Hamiltonian saddle node bifurcation x¨+ µx+ x2 = 0 (10.6.28) with the addition of higher order terms and forcing: x¨+ µx+ x2 + h.o.t. = δf(t). (10.6.29) The phase portrait of (10.6.28) is shown in Figure 10.6.1. The system (10.6.28) is Hamiltonian with H(x, p) = 1 2 p2 + 1 2 µx2 + 1 3 x3. (10.6.30) Let us first consider the system without higher order terms: x¨+ µx+ x2 = δf(t). (10.6.31) To study it, we rescale to blow up the singularity: x(t) = λξ(t) (10.6.32) where λ = ‖µ‖ and t = t√λ. We get ξ¨ − ξ + ξ2 = δ µ2 f ( τ√−µ ) , µ < 0, ξ¨ − ξ + ξ2 = δ µ2 f ( τ√ µ ) , µ > 0. (10.6.33) 216 10. Hamiltonian Bifurcation x x Figure 10.6.1. The evolution of the phase portrait of (10.6.27) as µ increases. The exponentially small estimates of Holmes, Marsden, and Scheurle [1988] apply to (10.6.33). One gets upper and lower estimates in certain algebraic sectors of the (δ, µ) plane. Now we consider x¨+ µx+ x2 + x3 = δf(t). (10.6.34) With δ = 0, there are equilibria at x = 0, −r, or − µ r and x˙ = 0, (10.6.35) where r = 1 + √ 1− 4µ 2 , (10.6.36) which is approximately 1 when µ ≈ 0. The phase portrait of (10.6.34) with δ = 0 and µ = − 12 is shown in Figure 10.6.2. As µ passes through 0, the small lobe undergoes the same bifurcation as in Figure 10.6.1, with the large lobe changing only slightly. Again we rescale by (10.6.36) to give ξ¨ − ξ + ξ2 − µξ3 = δ µ2 f ( τ√−µ ) , µ < 0, ξ¨ − ξ + ξ2 + µξ3 = δ µ2 f ( τ√ µ ) , µ > 0. (10.6.37) Notice that for δ = 0, the phase portrait is µ-dependent. The homoclinic orbit surrounding the small lobe for µ < 0 is given explicitly in terms of ξ by ξ(τ) = 4eτ( eτ + 23 )2 − 2µ, (10.6.38) 10.7 The Role of Dissipation 217 x Figure 10.6.2. The phase portrait of 10.6.31 with δ = 0. which is µ-dependent. An interesting technicality is that without the cubic term, we get µ-independent double poles at τ = ±ipi + log 2− log 3 in the complex τ -plane, while (10.6.38) has a pair of simple poles that splits these double poles to the pairs of simple poles at τ = ±ipi + log ( 2 3 ± i √ 2λ ) (10.6.39) where again λ = ‖µ‖. (There is no particular significance to the real part, such as log 2− log 3 in the case of no cubic term; this can always be gotten rid of by a shift in the base point ξ(0).) If a quartic term x4 is added, these pairs of simple poles will split into quartets of branch points and so on. Thus, while the analysis of higher order terms has this interesting µ-dependence, it seems that the basic exponential part of the estimates, exp ( − pi√‖µ‖ ) , (10.6.40) remains intact. 10.7 The Role of Dissipation If ze is an equilibrium point of a Hamiltonian vector field XH , then there are two methodologies for studying stability, as we already saw in the in- troductory chapter. (a) Energetics — determine if δ2H(ze) = Q is a definite quadratic form (Lagrange–Dirichlet). 218 10. Hamiltonian Bifurcation (b) Spectral methods — determine if the spectrum of the linearized oper- ator dXH(ze) = L is on the imaginary axis. The energetics method can, via reduction, be applied to relative equilib- ria too and is the basis of the energy–momentum method that we studied in Chapter 5. For general (not necessarily Hamiltonian) vector fields, the classical Li- apunov theorem states that if the spectrum of the linearized equations lies strictly in the left half plane, then the equilibrium is stable and even asymp- totically stable (trajectories starting close to the equilibrium converge to it exponentially as t → ∞). Also, if any eigenvalue is in the strict right half plane, the equilibrium is unstable. This result, however, cannot apply to the purely Hamiltonian case since the spectrum of L is invariant under re- flection in the real and imaginary coordinate axes. Thus, the only possible spectral configuration for a stable point of a Hamiltonian system is if the spectrum is on the imaginary axis. The relation between a and b is, in general, complicated, but one can make some useful elementary observations. Remarks. 1. Definiteness of Q implies spectral stability (i.e., the spectrum of L is on the imaginary axis). This is because spectral instability implies (linear and nonlinear) instability, while definiteness of Q implies sta- bility. 2. Spectral stability need not imply stability, even linear stability. This is shown by the unstable linear system q˙ = p, p˙ = 0 with a pair of eigenvalues at zero. 3. If Q has odd index (an odd number of negative eigenvalues), then L has a real positive eigenvalue; see Oh [1987]. Indeed, in canonical coordinates, and identifyingQ with its corresponding matrix, we have L = JQ. Thus, detL = detQ is negative. Since detL is the product of the eigenvalues of L and they come in conjugate pairs, there must be at least one real pair of eigenvalues, and in fact an odd number of positive real eigenvalues. 4. If P = T ∗Q with the standard symplectic structure (no magnetic terms) and if H is of the form kinetic plus potential so that an equi- librium has the form (qe, 0), and if δ2V (qe) has negative index, then again L must have real eigenvalues. This is because one can diago- nalize δ2V (qe) with respect to the kinetic energy inner product, in which case the eigenvalues are evident. 10.7 The Role of Dissipation 219 To get more interesting effects than those covered by the above remarks, we consider gyroscopic systems; that is, linear systems of the form Mq¨ + Sq˙ + Λq = 0 (10.7.1) where M is a positive definite symmetric n × n matrix, S is skew, and Λ is symmetric. The term with S is the gyroscopic, or magnetic term. As we observed earlier, this system is verified to be Hamiltonian with p = Mq˙, energy function H(q, p) = 1 2 pM−1p+ 1 2 qΛq (10.7.2) and the bracket {F,K} = ∂F ∂qi ∂K ∂pi − ∂K ∂qi ∂F ∂pi − Sij ∂F ∂pi ∂K ∂pj . (10.7.3) If the index of V is even (see Remark 3) one can get situations where δ2H is indefinite and yet spectrally stable. Roughly, this is a situation that is capable of undergoing a Hamiltonian Hopf bifurcation, so includes examples like the “cowboy” solution for the double spherical pendulum and certain regimes of the heavy top. 10.7.1 Theorem (Dissipation induced instabilities). If the system (10.7.1) is modified to Mq¨ + (S + R)q˙ + Λq = 0 (10.7.4) for small > 0 and R symmetric and positive definite, then the perturbed linearized equations z˙ = Lz are spectrally unstable, that is, at least one pair of eigenvalues of L is in the right half plane. This result, due to Bloch, Krishnaprasad, Marsden, and Ratiu [1994] builds on basic work of Chetayev [1961, 1989] and Hahn [1967]. The argu- ment proceeds in two steps. Step 1. Construct the Chetayev functional W (q, p) = H(q, p) + βp · (Λq) (10.7.5) for small β. This function has the beautiful property that for β small enough, W has the same index as H, yet W˙ is negative definite, where the overdot is taken in the dynamics of (10.7.4). This alone is enough to prove Liapunov instability, as is seen by studying the equation W (q(T ), p(T )) = W (q0, p0) + ∫ T 0 W˙ (q(t), p(t))dt (10.7.6) and choosing (q0, p0) in the sector where W is negative, but arbitrarily close to the origin. 220 10. Hamiltonian Bifurcation Step 2. Employing an argument of Hahn [1967] to show spectral insta- bility. Here one uses the fact that is small and the original system is Hamilto- nian. Indeed, the only nontrivial possibility to exclude for the eigenvalues on the imaginary axis is that they all stay there and are not zero for 6= 0. Indeed, they cannot all move left by Step 1 and L cannot have zero eigen- values since Lz = 0 implies W˙ (z, z) = 0. However, in this case, Hahn [1967] shows the existence of at least one periodic orbit, which cannot exist in view of (10.7.6) and the fact that W˙ is negative definite. This argument generalizes in two significant ways. First, it is valid in infinite dimensional systems, where M , S, R, and Λ are replaced by linear operators. One of course needs some technical conditions to ensure that W has the requisite properties and that the evolution equations generate a semi-group on an appropriate Banach space. For Step 2 one requires, for example, that the spectrum at = 0 be discrete with all eigenvalues having finite multiplicity. The second generalization is to systems in block diagonal form but with a non-Abelian group. The system (10.7.4) is the form that block diagonal- ization gives with an Abelian symmetry group. For a non-Abelian group, one gets, roughly speaking, a system consisting of (10.7.4) coupled with a Lie–Poisson (generalized rigid body) system. The main step needed in this case is a generalization of the Chetayev functional. This formulation is attractive because of the interesting conclusions that can be obtained essentially from energetics alone. If one is willing to make additional assumptions, then there is a formula giving the amount by which simple eigenvalues move off the imaginary axis. One version of this formula, due to MacKay [1991] states that Reλ = ξ¯(JB)antiξ ξ¯T Jξ +O(2) (10.7.7) where we write the linearized equations in the form z˙ = Lz = (JQ+ B)z. (10.7.8) λ is the perturbed eigenvalue associated with a simple eigenvalue λ0 = iω0 on the imaginary axis at = 0, ξ is a (complex) eigenvector for L0 with eigenvalue λ0, and (JB)anti is the antisymmetric part of JB. In fact, the ratio of quadratic functions in (10.7.7) can be replaced by a ratio involving energy-like functions and their time derivatives including the energy itself or the Chetayev function. To actually work out (10.7.7) for examples like (10.7.1) can involve considerable calculation. Here is a simple example in which one can carry out the entire analysis directly. We hasten to add that problems like the double spherical pendu- lum are considerably more complex algebraically and a direct analysis of the eigenvalue movement would not be so simple. 10.7 The Role of Dissipation 221 Consider the system (see Chetayev [1961, 1989] ) x¨− gy˙ + γx˙+ αx = 0, y¨ + gx˙+ δy˙ + βy = 0, (10.7.9) which is a special case of (10.7.4). Assume γ ≥ 0 and δ ≥ 0. For γ = δ = 0 this system is Hamiltonian with symplectic form Ω = dx ∧ dx˙+ dy ∧ dy˙ − gdx ∧ dy (10.7.10) and Hamiltonian H = 1 2 (x˙2 + y˙2) + 1 2 (αx2 + βy2). (10.7.11) (Note that for α = β, angular momentum is conserved.) The characteristic polynomial is computed to be p(λ) = λ4 + (γ+ δ)λ3 + (g2 +α+ β+ γδ)λ2 + (γβ+ δα)λ+αβ. (10.7.12) Let p0(λ) = λ4 + (g2 + α+ β)λ2 + αβ. (10.7.13) Since p0 is a quadratic in λ2, its roots are easily found. One gets: (i) If α > 0, β > 0, then H is positive definite and the eigenvalues are on the imaginary axis; they are coincident in a one-to-one resonance for α = β. (ii) If α and β have opposite signs, then H has index 1 and there is one eigenvalue pair on the real axis and one pair on the imaginary axis. (iii) If α < 0 and β < 0 then H has index 2. Here the eigenvalues may or may not be on the imaginary axis. To determine the cases, let D = (g2 + α+ β)2 − 4αβ = g4 + 2g2(α+ β) + (α− β)2 be the discriminant. Then the roots of (10.7.13) are given by λ2 = 1 2 [−(g2 + α+ β)± √ D]. Thus we arrive at (a) If D < 0, then there are two roots in the right half plane and two in the left. (b) If D = 0 and g2 + α + β > 0, there are coincident roots on the imaginary axis, and if g2 + α + β < 0, there are coincident roots on the real axis. 222 10. Hamiltonian Bifurcation (c) If D > 0 and g2 +α+β > 0, the roots are on the imaginary axis and if g2 + α+ β < 0, they are on the real axis. Thus the case in which D ≥ 0 and g2+α+β > 0 (i.e., if g2+α+β ≥ 2√αβ), is one to which the dissipation induced instabilities theorem applies. Note that for g2 +α+β > 0, if D decreases through zero, a Hamiltonian Hopf bifurcation occurs. For example, as g increases and the eigenvalues move onto the imaginary axis, one speaks of the process as gyroscopic stabilization . Now we add damping and get 10.7.2 Proposition. If α < 0, β < 0, D > 0, g2 + α + β > 0 and least one of γ, δ is strictly positive, then for (10.7.9), there is exactly one pair of eigenvalues in the strict right half plane. Proof. We use the Routh–Hurwitz criterion (see Gantmacher [1959, Vol- ume 2]), which states that the number of strict right half plane roots of the polynomial λ4 + ρ1λ3 + ρ2λ2 + ρ3λ+ ρ4 equals the number of sign changes in the sequence{ 1, ρ1, ρ1ρ2 − ρ3 ρ1 , ρ3(ρ1ρ2 − ρ3)− ρ1ρ4 ρ1ρ2 − ρ3 , ρ4 } . (10.7.14) For our case, ρ1 = γ + δ > 0, ρ2 = g2 + α+ β + γδ > 0, ρ3 = γβ + αδ < 0, and ρ4 = αβ > 0, so the sign sequence (10.7.14) is {+,+,+,−,+}. Thus, there are two roots in the right half plane. It is interesting to speculate on the effect of damping on the Hamiltonian Hopf bifurcation in view of these general results and in particular, this example. For instance, suppose g2 + α + β > 0 and we allow D to increase so a Hamiltonian Hopf bifurcation occurs in the undamped system. Then the above sign sequence does not change, so no bifurcation occurs in the damped system; the system is unstable and the Hamiltonian Hopf bifurca- tion just enhances it. However, if we simulate forcing or control by allowing one of γ or δ to be negative, but still small, then the sign sequence is more complex and one can get, for example, the Hamiltonian Hopf bifurcation breaking up into two nearly coincident Hopf bifurcations. This is consistent with the results of van Gils, Krupa, and Langford [1990]. 10.8 Double Bracket Dissipation 223 10.8 Double Bracket Dissipation Bloch, Krishnaprasad, Marsden, and Ratiu [1996] study the phenomenon of dissipation induced instabilities for Euler–Poincare´ systems on Lie alge- bras or equivalently, for Lie–Poisson systems on the duals of Lie algebras. In the previous section, we indicated that if a mechanical system with sym- metry has an indefinite second variation of the augmented Hamiltonian at a relative equilibrium, as determined by the energy–momentum method, then the system becomes spectrally unstable with the addition of a small amount of dissipation. That dissipation was of Rayleigh dissipation type, and was added to the internal variables of the system and the methods that were used to prove this were essentially those of linear analysis. For systems on Lie algebras, or equivalently, for invariant systems on Lie groups, one cannot have linear dissipative terms of Rayleigh dissipa- tion type in the equations in the naive sense. However, when restricted to coadjoint orbits, these dissipation terms can be obtained from a gradient structure that is similar in spirit to the way one gets dissipative terms from the gradient of a Rayleigh dissipation function. In this context, one gets dissipation induced instabilities, similar to what one has in the case of in- ternal dissipation. This means that the addition of dissipation to a state that is not formally stable forces at least one pair of eigenvalues into the right half plane, which of course implies nonlinear instability. One of the interesting features is the method of construction of the non- linear dissipative terms. This is done by utilizing the double bracket equa- tion of Brockett (see Brockett [1988, 1993]) to the present context. In fact, this form is well adapted to the study of dissipation on Lie groups since it was originally constructed as a gradient system and it is well known in other contexts that this formalism plays an important role in the study of integrable systems (see, for example, Bloch, Flaschka, and Ratiu [1990] and Bloch, Brockett, and Ratiu [1992]). We will also show that this type of dissipation can be described in terms of a symmetric Poisson bracket. Symmetric brackets for dissipative sys- tems have been considered by Kaufman and Dewar [1984], Kaufman [1985], Grmela [1984a, 1984b, 1986], Morrison [1986]. It is not clear how the brack- ets of the present paper are related to those. Our brackets are more directly motivated by those in Vallis, Carnevale, and Young [1989], Shepherd [1992] and references therein. We present a class of symmetric brackets that are systematically con- structed in a general Lie algebraic context. We hope that our construction might shed light on possible general properties that these brackets might have. The general equations of motion that we consider have the following form: F˙ = {F,H}skew − {F,H}sym 224 10. Hamiltonian Bifurcation where H is the total energy. In many cases however, especially those in- volving thermodynamics, one replaces H in the second bracket by S, the entropy. We refer to the above references for this aspect; it remains for the future to link that work more closely with the present context and to see in what sense, if any, the combined bracket satisfies a graded form of Jacobi’s identity. It is interesting that the type of dissipation described here is of con- siderable physical interest. For example, as we shall point out below, the Landau–Lifschitz (or Gilbert) dissipative mechanism in ferromagnetics is exactly of the type we describe and this dissipative mechanism is well ac- cepted and studied (see O’Dell [1981] for example). In geophysical situa- tions, one would like a dissipative mechanism that separates the different time scales of decay of the energy and the enstrophy. That is, one would like a dissipative mechanism for which the energy decays but the enstro- phy remains preserved. This is exactly the sort of dissipative mechanism described here and that was described in Vallis, Carnevale, and Young [1989], Shepherd [1992] and references therein. Also, in plasma physics and stellar dynamics, one would like to have a dissipative mechanism that pre- serves the underlying conservation of particle number, yet has energy decay. Again, the general mechanism here satisfies these properties (see Kandrup [1991] and Kandrup and Morrison [1992]). We will discuss all of these ex- amples in the body of the paper. To get a concrete idea of the type of dissipative mechanism we have in mind, we now give a simple example of it for perhaps the most basic of Euler–Poincare´, or Lie–Poisson systems, namely the rigid body. Here, the Lie algebra in question is that of the rotation group; that is, Euclidean three space R3 interpreted as the space of body angular velocities Ω equipped with the cross product as the Lie bracket. On this space, we put the stan- dard kinetic energy Lagrangian L(Ω) = 12 (IΩ) · Ω (where I is the moment of inertia tensor) so that the general Euler–Poincare´ equations (discussed below in §4) become the standard rigid body equations for a freely spinning What does §4 refer to?rigid body: IΩ˙ = (IΩ)× Ω, (10.8.1) or, in terms of the body angular momentum M = IΩ, M˙ = M × Ω. In this case, the energy equals the Lagrangian; E(Ω) = L(Ω) and energy is conserved by the solutions of (10.8.1). Now we modify the equations by adding a term cubic in the angular velocity: M˙ = M × Ω + αM × (M × Ω), (10.8.2) where α is a positive constant. A related example is the 1935 Landau-Lifschitz equations for the mag- netization vector M in a given magnetic field B (see, for example, O’Dell 10.8 Double Bracket Dissipation 225 [1981, p. 41]): M˙ = γM ×B + λ‖M‖2 (M × (M ×B)), (10.8.3) where γ is the magneto-mechanical ratio (so that γ‖B‖ is the Larmour frequency) and λ is the damping coefficient due to domain walls. In each case, it is well known that the equations without damping can be written in either Euler–Poincare´ form or in Lie–Poisson (Hamiltonian) form. The equations are Hamiltonian with the rigid body Poisson bracket: {F,K}rb(M) = M · [∇F (M)×∇K(M)] with Hamiltonians given respectively by H(M) = (M · Ω)/2 and H(M) = γM ·B. One checks in each case that the addition of the dissipative term has a number of interesting properties. First of all, this dissipation is derivable from an SO(3)-invariant force field, but it is not induced by any Rayleigh dissipation function in the literal sense. However, it is induced by a Rayleigh dissipation function in the following restricted sense: It is a gradient when restricted to each momentum sphere (coadjoint orbit) and each sphere car- ries a special metric (later to be called the normal metric). Namely, the extra dissipative term in (10.8.2) equals the negative gradient of the Hamil- tonian with respect to the following metric on the sphere. Take a vector v in R3 and orthogonally decompose it in the standard metric on R3 into components tangent to the sphere ‖M‖2 = c2 and vectors orthogonal to this sphere: v = M · v c2 M − 1 c2 [M × (M × v)]. (10.8.4) The metric on the sphere is chosen to be ‖M‖−2α times the standard inner product of the components tangent to the sphere in the case of the rigid body model and just λ times the standard metric in the case of the Landau–Lifschitz equations. Secondly, the dissipation added to the equations has the obvious form of a repeated Lie bracket, that is, a double bracket, and it has the properties that the conservation law d dt ‖M‖2 = 0 (10.8.5) is preserved by the dissipation (since the extra force is orthogonal to M) and the energy is strictly monotone except at relative equilibria. In fact, we have d dt E = −α‖M × Ω‖2, (10.8.6) for the rigid body and d dt E = − λ‖M‖2 ‖M ×B‖ 2, (10.8.7) 226 10. Hamiltonian Bifurcation in the case of the Landau–Lifschitz equations, so that trajectories on the angular momentum sphere converge to the minimum (for α and λ positive) of the energy restricted to the sphere, apart from the set of measure zero consisting of orbits that are relative equilibria or are the stable manifolds of the perturbed saddle point. Another interesting feature of these dissipation terms is that they can be derived from a symmetric bracket in much the same way that the Hamil- tonian equations can be derived from a skew symmetric Poisson bracket. For the case of the rigid body, this bracket is {F,K} = α(M ×∇F ) · (M ×∇K). (10.8.8) As we have already indicated, the same formalism can be applied to other systems as well. In fact, later in the paper we develop an abstract construction for dissipative terms with the same general properties as the above examples. When this method is applied to fluids one gets a dissipa- tive mechanism related to that of Vallis, Carnevale, and Young [1989] and Shepherd [1992] as follows. One modifies the Euler equations for a perfect fluid, namely ∂v ∂t + v · ∇v = −∇p (10.8.9) where v is the velocity field, assumed divergence free and parallel to the boundary of the fluid container, and where p is the pressure. With dissipa- tion, the equations become: ∂v ∂t + v · ∇v = −∇p+ αP (( £u(v)v[ )]) (10.8.10) where α is a positive constant, P is the Hodge projection onto the diver- gence free part, and where u(v) = P (( £vv[ )]) . The flat and sharp symbols denote the index lowering and raising operators induced by the metric; that is, the operators that convert vectors to one forms and vice versa. Written in terms of the vorticity, these equations become d dt ω + £vω = α£u(v)ω. 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Page 246 Index A absolutely Γ-irreducible, 202 acceleration feedback, 137 action principle, 33 action tensor, 51 adjoint action, 40, 95 admissible variations, 17 admits horseshoes, 209 algorithm, 173 algorithms for the rigid body, 188 amended potential, 18, 60 angular momentum, 142 angular velocity, 56 antisymplectic symmetries, 151 Arnold diffusion, 213 Arnold form, 110 attitude controllability, 145 attitude drift, 141 attitude equation, 142 augmented Hamiltonian, 17, 86 averaging, 133 B ball in a rotating hoop, 192 bead, 19 bead on the rotating hoop, 132 beyond all orders, 214 bifurcation theory, 191 block diagonalization, 18, 109 block diagonalization theorem, 111 blow up, 215 body angular momentum, 9 body angular velocity, 6 book, 22 C canonical 1-form, 32 canonical 2-form, 32 canonical cotangent coordinates, 32 canonical Poisson bracket, 12 Carnot–Caratheodory metrics, 150 Cartan connection, 132 Cartan involutions, 153 Cartan structure equation, 60 Casimir function, 16, 36 cat, 23 Cayley transform, 175 center of mass, 54 chaotic, 11, 208 Index 247 charge, 147 charged particle, 145 Chetayev functional, 219 circular drop, 201 classical commutation relations, 40 classical mechanical system, 5 classical water molecule, 152 coadjoint action, 40 complex duals, 203 configuration space, 5 connection, 47, 124, 125 conservation of energy, 31 consistent, 173 constraint forces, 194 control manouvers, 144 control system, 137, 148 controls, 145 convergence, 174 convexity theorem, 45 cotangent bundle, 5, 32 cotangent bundles, 159 cotangent relative equilibria, 88 coupled rigid bodies, 51, 153, 194 coupling, 79 covariant exterior derivative, 60 “cowboy” solution, 219 crank, 24 critical group orbit, 86 critical rotation rate, 197 current configuration, 6 curvature, 60, 62 cusp, 197 cusp catastrophe, 197 D d’Alembert, 72 d’Alembert principle, 140 Darboux’ Theorem, 32 discrete reduction, 156 discrete reduction procedure, 156 discrete symmetries, 113 discrete symmetry group, 151 discrete symmetry of the water molecule, 167 dissipation, 217 dissipation induced instabilities, 219 double spherical pendulum, 162 double bracket, 223 double spherical pendulum, 3, 51, 67, 91, 117, 152, 156 double whirling mass system, 152 drop, 151, 193, 201 dual pair, 171 dual pairs, 168 dual spin, 137, 139 dynamic phase, 21, 125, 129 E efficient, 23 eigenvalue, 206 eigenvector, 206 Einstein’s equations, 206 elimination of the node, 44 energetics, 200, 217 energy preserving algorithm, 175 energy–Casimir method, 16, 103 energy–momentum method, 17, 102, 165 energy–momentum theorem, 102 energy-integrator, 174 equations of motion, 5 equilibrium solutions, 197 equivariant, 40 Euler angles, 6 Euler equations, 35 Euler–Lagrange equations, 33, 73 explicit symplectic scheme, 174 exponentially small, 214 F falling cat, 22, 148, 153 feedback law, 136 first variation equation, 15 fixed point set, 155, 158 flexible appendages, 144 flow, 31 fluid, 51 forced beam, 212 Foucault pendulum, 19 free boundary fluids, 186 248 Index free spinning rotor, 139 friction, 194 G Γ-irreducible, 202 gauge invariance, 33, 104 general relativity, 206 generating functions, 179 generic point, 101 geometric phase, 19, 21, 125, 129 geometric phases, 182 gravitating fluid masses, 16 gyroscopic, 72 gyroscopic form, 115, 136 gyroscopic instability, 194 gyroscopic Lagrangians, 141 gyroscopic systems, 219 H Hamilton’s equations, 12, 30 Hamilton’s principle, 33 Hamilton–Jacobi theory, 179 Hamiltonian, 5, 34 Hamiltonian bifurcation, 191 Hamiltonian form, 5 Hamiltonian Hopf bifurcation, 194, 199 Hamiltonian pitchfork, 192 Hamiltonian pitchfork bifurcation, 197 Hamiltonian saddle node bifurca- tion, 215 Hamiltonian vector field, 30 Hannay–Berry connection, 132 Hannay-Berry phase, 134 harmonic oscillators, 171 hinge angle, 121 holonomy, 19, 122 holonomy drive, 24 holonomy of a rigid body, 22 homoclinic chaos, 208 homoclinic orbit, 22 hoop, 132, 192 Hopf fibration, 21, 26, 47 Hopf map, 171 horizontal lift, 124 horizontal space, 57 horizontal-vertical decomposition, 59 Hyperion, 11 hyperregular, 34 I ideal fluids, 51 implicit symplectic scheme, 175 independent passing, 202 inertia tensor, 97, 135 inertial frame, 134 integrable Hamiltonian systems, 168 internal modes, 19 internal-rigid interaction terms, 116 isotropy, 191 isotypic components, 157 J Jacobi–Bertrand–Haretu coordinates, 53 Jacobi-Liouville theorem, 44 K Kaluza–Klein configuration space, 146 Kaluza–Klein description, 145 Kaluza–Klein Lagrangian, 146 Kaluza-Klein construction, 75 kinetic energy, 6, 33 Kowalewski top, 153, 156 Krein collision, 194 L Lagrange-Dirichlet Criterion, 16 Lagrangian, 33 Lagrangian reduction, 72 Legendre transform, 34 Leggett equations, 211 Liapunov’s theorem, 15 Lie–Poisson bracket, 13, 35 Lie–Poisson Hamilton–Jacobi equa- tion, 184 Index 249 Lie–Poisson Reduction, 42 Lie–Poisson reduction, 35, 184 Lie–Trotter formula, 174 limitations on mechanical integra- tors, 177 linearized oscillations, 197 liquid drop, 151, 167 locked inertia tensor, 18, 56, 135 Lorentz force law, 46 M magnetic field, 46, 145, 146 magnetic resonance, 24 magnetic term, 62 magnetic terms, 147 magnetic two form, 72 mechanical connection, 18, 57 mechanical integrator, 173 Melnikov method, 207 Melnikov vector, 213 Melnikov’s method, 11 micromotors, 24 microorganisms, 24 mid point rule, 175 moment of inertia tensor, 7 momentum map, 14, 39, 45 momentum maps, 36 momentum-integrator, 174 Moncrief splitting, 112 monitor accuracy, 179 monopole, 47 moving systems, 130 N Newton’s second law, 194 no-twist, 153 Noether’s theorem, 39 non-canonical bracket, 13 non-integrable, 11 nonsymmetric variations, 167 normal form, 109, 114 normalizer, 155 O One-to-one resonance, 199 one-to-one resonance, 161, 171, 194 optimal control, 23, 148, 153 orbit reduction theorem, 44 orbit symplectic structure, 43 P parallel transport, 19 particle in a magnetic field, 47 particle in a Yang–Mills, 145 passing, 200 pendulum equation, 196 phase shift, 141 phase space, 5 pitchfork, 197 planar coupled rigid bodies, 193 planar double pendulum, 162 planar liquid drop, 193 planar rigid bodies, 121 Poisson, 30 Poisson action, 39 Poisson bracket, 30 Poisson manifold, 29 Poisson reduction, 42 Poisson tensor, 50 potential energy, 33 principal axes, 7 Q quadratic singularities, 45, 206 quaternionic, 203 R reconstruction, 123, 124 reduced energy–momentum method, 113 reduced Hamilton–Jacobi equation, 184 reduced Hamiltonian, 64 reduced Hamiltonian system, 63 reduced phase space, 10 reduced principal bundle, 82 reduced space, 61 reduced system, 14 reduced variational principle, 76 reduction by stages, 63 250 Index Reduction theorem, 43 reduction theorem, 14 reference configuration, 6, 96 regular point, 101 regular quadratic Lie algebra, 186 regularized, 206 relative equilibria, 10, 14 relative equilibrium, 17, 85 relative equilibrium theorem, 85 rigid bodies, 51 rigid body, 16, 21, 95, 135, 164, 187 rigid body bracket, 12 rigid body phases, 128 rigid body-rotor system, 141 rigid variations, 18 rigid-internal splitting, 109 rigid-rigid terms, 116 robot, 25 rotating earth, 131 rotating liquid drop, 151, 201 rotating waves, 86 rotation group, 26 rotational invariance, 13 rotors, 135 Routh-Hurwitz criterion, 222 Routhian, 72, 141 S saddle node bifurcation, 215 satellite parking, 144 semi-direct product, 155 semi-direct product reduction the- orem, 63 shape, 91 shape space, 23, 49, 62, 121 shape variations, 18 shifting construction, 45 shifting map, 59 simple mechanical G-system, 50 simple pendulum, 140, 176 singular points, 206 singularities, 45 solution symmetry breaking, 193 space telescope, 25 spatial angular momentum, 13 spatial angular velocity, 6 spectral methods, 218 spectral stability, 218 spectroscopy, 25 spherical pendulum, 51, 65, 90, 152, 156, 161 splitting, 199 stability, 15, 174 stabilization, 135, 140 standard map, 176 steady rotations, 10 steady state bifurcation, 193 steady-state bifurcation, 199 step size, 173 structure constants, 35 subblocking, 113, 157 subblocking property, 166 subblocking theorem, 165 superfluid 3He, 211 surface tension, 151 suspended system, 209 symmetric symplectic algorithms, 180 symmetric variations, 166 symmetry breaking parameter, 197 symmetry groups, 192 symplectic action, 42 symplectic form, 13 symplectic leaves, 33 symplectic manifold, 30 symplectic map, 13 symplectic reduction, 42 symplectic submanifold, 156 symplectic-integrator, 174 system parameters, 91 system symmetry breaking, 192 T tangent coordinates, 33 The Poincare´–Melnikov method, 207 theta functions, 21 three body problem, 1 three-body problem, 214 Index 251 time dependent inertia tensor, 97 topologically non-trivial, 26 transversal intersecting separatri- ces, 208 transversality property, 158 U unit circle bundle, 26 V variational principle, 72 velocity shift, 140 vortex algorithms, 186 vortex dipoles, 186 W water molecule, 1, 52, 69, 93, 166 whirling masses, 162 whirling planar pendulum, 196 Wong kinetic term, 75 Wong’s equations, 23 Y Yang–Mills equations, 207 Yang–Mills field, 24 Yang–Mills particles, 148

Lectures on mechanics

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