Geometric Averaging of Hamiltonian Systems: Periodic Solutions, Stability, and KAM Tori

We investigate the dynamics of various problems defined by Hamiltonian systems of two and three degrees of freedom that have in common that they can be reduced by an axial symmetry. Specifically, the systems are either invariant under rotation about the vertical axis or can be made approximately axially symmetric after an averaging process and the corresponding truncation of higher-order terms. Once the systems are reduced we study the existence and stability of relative equilibria on the reduced spaces which are unbounded two- or four-dimensional symplectic manifolds with singular points. We establish the connections between the existence and stability of relative equilibria and the existence and stability of families of periodic solutions of the full problem. We also discuss the existence of KAM tori surrounding the periodic solutions.