Nearly-integrable perturbations of the Lagrange top: applications of kam theory

Abstract: Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple) normal 1: −1 resonance. This theory guarantees the persistence of the invariant torus in the Diophantine case and makes possible a further quasi-periodic normal form, necessary for investigation of the non-linear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation. 1. The Lagrange top The Lagrange top is an axially symmetric rigid body in a three dimensional space, subject to a constant gravitational field such that the base point of the body-symmetry (or figure) axis is fixed in space, see Figure 1. Mathematically speak-ing, this is a Hamiltonian system on the tangent bundle TSO(3) of the rota-tion group SO(3) with the symplectic 2-form σ. This σ is the pull-back of the canonical 2-form on the co-tangent bundle T ∗SO(3) by the bundle isomorphis