Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups

This paper presents theorems which establish the existence of horseshoes and Arnold diffusion for nearly integrable Hamiltonian systems associated with Lie groups. The methods are based on our two previous papers, Holmes and Marsden [1982a], [1982b). The two main examples treated here are as follows: 1. A simplified model of the rigid body with attachments. This system has horseshoes (with one attachment) and Arnold diffusion (with two or more attachments) . 2. A rigid body under gravity, close to a symmetric (Lagrange) top. This system is shown to have horseshoes (and hence is not integrable). The main new feature here is the presence of Lie groups. Both the symmetry groups and the basic phase spaces involve Lie groups and our perturbation methods must be modified to take this into account. As in our previous work, the results hinge on reduction together with a method of Melnikov. This is used to analyze the perturbation of a homoclinic orbit in an integrable Hamiltonian system. In the first example the unperturbed system is the free rigid body which has a homoclinic orbit lying on a sphere. This sphere arises as the coadjoint orbit for the rotation group SO (3), and the computation of Poisson brackets needed in the Melnikov theory is most easily done using the (Kirillov, Arnold, Kostant and Souriau) theory of coadjqint orbits and the Lie-Poisson bracket on the dual of a Lie algebra. This theory is-reviewed in Section 2. Reduction in the sense of Marsden and Weinstein [1974] shows that the phase space for a rigid body under gravity is T* S2, the cotangent bundle of a sphere. This and its connection with Euler angles and coadjoint orbits in the Euclidean group is explained in Section 3. This section thus sets up the basic phase spaces needed in the analysis of our second example. Section 4 develops the Melnikov theory when the phase space is a product of the dual of a Lie algebra and a set of action angles variables. This is applied to a model problem based on the rigid body with attachments in Section 5. Section 6 develops the Melnikov theory for systems on a phase space where the unperturbed system admits an Sl symmetry and has a homo clinic orbit in the