Variational methods and periodic solutions of N-body and N-centre problems.

In this thesis we study periodic solutions of several N-body and N-centre systems with different potentials from a variational viewpoint. The underlying focus is on understanding the structure of various action functionals, and the relationship between this and the system's periodic orbits and their properties. In particular we: investigate the integrable central force problems with potentials Valpha(x) = - 1/|x|alpha for 1≤ alpha ≤ 2. We show that for 1 0 we list the finitely many distinct critical manifolds of collisionless orbits in that class in order of their action, and label them with their Morse indices with respect to the action functional. We investigate the 2-body/1-centre problem with Lennard-Jones potential. We find the region of energy-momentum space that supports the existence of periodic or quasiperiodic motion. We also show there exists m ∈ N such that for all q ≥ m there are periodic orbits in Op= UT>0 OpT (p ∈ Z - {0}) with q 'radial oscillations' in one period. Obtain results on which homotopy and homology classes of loops contain periodic solutions of the symmetric planar Newtonian 2-centre problem, see theorem 4.4.1. In particular we find that the integrability of the system places strong constraints on which homology classes of loops contain periodic solutions and obtain some interesting results on prime-period solutions. We also order by action 'P1 orbits' in all homotopy classes of loops that contain them and label them by their Morse indices with respect to the action functional. We investigate the 7V-body problem with identical particles interacting through a potential of Lennard-Jones (LJ) type. We consider any subset of loop space that satisfies a few basic conditions, one of which corresponds to the notion of 'tiedness' introduced by Gordon in [41]. We show that this system admits periodic solutions in every homotopy class of this subset of loop space. More precisely we show that every homotopy class contains at least two periodic solutions for sufficiently large periods. One of these solutions is a local minimum and the other is a mountain-pass critical point of the action functional. We also prove that given a homotopy class of one of these subsets there do not exist any periodic solutions in it for sufficiently small periods. The results have wide applicability. For example, one can consider the space of choreographies and prove existence results for choreographical solutions. Our existence proof relies upon an assumption that global minimizers of standard strong force potentials on suitable spaces are nondegenerate up to some symmetries. We also find periodic solutions in some classes of loops that do not satisfy any tiedness condition. In particular we use a result in [27] to construct a periodic solution of the restricted spatial (2N + 2)-body problem. o present a note on McCord, Montaldi, Roberts and Sbano's paper on relative periodic orbits in symmetric Lagrangian systems, see [54]. For the N-centre problem with a strong force potential that is bounded above we find those homotopy classes of relative loops on which the action functional is coercive. We describe the homotopy types of the homotopy classes of relative loops. Under the assumption that the action functional is an S1-invariant Morse function we describe a set of homotopy classes that contain infinitely many periodic orbits. The work can be viewed as an expansion on an example presented at the end of [54] which had N = 2; in particular we correct and generalize some assertions made regarding coercivity of the action functional and the centralizers of the g-twisted action. Some of the results are subject to numerically motivated assumptions detailed in the thesis