M. Golubitsky, J.-M. Mao and M. Nicol
Symmetries of Periodic Solutions for Planar Potential Systems
Proc. Amer. Math. Soc. 124 (1996) 3219-3228.
In this article we discuss the symmetries of periodic solutions to
two degree of freedom Hamiltonian systems in mechanical form. The
possible symmetries of such periodic trajectories are generated by
spatial symmetries (a finite subgroup of O(2)), phase-shift symmetries
(the circle group S^1), and a time-reversing symmetry (associated with
mechanical form). We focus on the symmetries and structures of the
trajectories in configuration space (R^2) showing
that special properties such as self-intersections and
brake orbits are consequences of symmetry.