| We establish a criterion on potential energy functions which, when satisfied, asserts the existence of an infinite number of periodic orbits in a dynamical system defined by two particles moving on the two-dimensional (or "flat") torus. The original system is reduced to that of a single point-mass moving about the torus, for which we find a continuum of trajectories satisfying a particular symmetry relation. Using a system of Poincare maps, we obtain addition information about a particular subset of these trajectories in order to describe their behaviour in a linear portion of the space. Finally, we show under certain additional assumptions that, for any sufficiently large two-dimensional torus, a countably infinite subset of these trajectories are periodic. |
