Symmetries of periodic solutions for planar potential systems

In this article we discuss the symmetries of periodic solutions to Hamiltonian systems with two degrees of freedom in mechanical form. The possible symmetries of such periodic trajectories are generated by spatial symmetries (a finite subgroup of $\text {{\bf O(2)}})$, phase-shift symmetries (the circle group $\text {{\bf S}}^1)$, and a time-reversing symmetry (associated with mechanical form). We focus on the symmetries and structures of the trajectories in configuration space ($\mathbb R^2$), showing that special properties such as self-intersections and brake orbits are consequences of symmetry.