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Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups

Abstract : The Melnikov theory of perturbations of Hamiltonian systems containing homoclinic orbits is extended to systems containing canonical variables belonging to the coadjoint orbits of a Lie group. This is applied to the free rigid body with attachments and to the nearly symmetric top. These systems are thereby shown to have transverse homoclinic manifolds in an appropriate return map and therefore have complex dynamics. In particular, the heavy top and rigid body with one attachment are shown to contain horseshoes and therefore have no additional analytic integrals, while the rigid body with several attachments exhibits Arnold diffusion.