SYMMETRIES OF PERIODIC-SOLUTIONS FOR PLANAR POTENTIAL SYSTEMS

In this article we discuss the symmetries of periodic solutions to Ham iltonian systems with two degrees of freedom in mechanical form. The p ossible symmetries of such periodic trajectories are generated by spat ial symmetries (a finite subgroup of O(2)), phase-shift symmetries (th e circle group S-1), and a time-reversing symmetry (associated with me chanical form). We focus on the symmetries and structures of the traje ctories in configuration space (R(2)), showing that special properties such as self-intersections and brake orbits are consequences of symme try.