A Minimax Approach to Relative Periodic Orbits in Symmetric Three-Degree-of-Freedom Hamiltonian Systems

Abstract

We study three-degree-of-freedom Hamiltonian systems that are invariant under rotations about the $z$-axis and under reflection across the $xy$-plane. Fixing the angular momentum, such systems reduce to Hamiltonian systems with two degrees of freedom. We focus on the range of energy values for which the corresponding Hill regions are compact. First, under suitable assumptions on the topology of these compact Hill regions, we prove the existence of periodic solutions on each prescribed energy surface of the reduced system by means of a variational minimax method. These periodic solutions are obtained as saddle points of the Maupertuis functional. The resulting solutions are either nontrivial spatial periodic solutions or trivial planar brake solutions in the reduced system. Next, by computing the Morse index, we provide a sufficient condition ensuring that the periodic solutions obtained are nontrivial. Finally, we apply our results to the isosceles three-body problem and to the spatial anisotropic Kepler problem. In both cases, we verify the sufficient condition for nontriviality and thereby establish the existence of nontrivial periodic solutions.