Periodic orbits of inelastic particles on a ring

We consider the dynamics of N rigid particles of arbitrary mass that are constrained to move on a frictionless ring. Collisions between particles are inelastic with a constant coefficient of restitution e, and between collisions the particles move with constant velocity. We study sequences of collisions that are self-similar in the sense that the relative positions return to their original relative positions after the collision sequence while the relative velocities are reduced by a constant factor. For a given collision sequence, we develop the analytic machinery to determine the particle velocities and the locations of collisions, and we show that the problem of determining self-similar orbits reduces to solving an eigenvalue problem to obtain the particle velocities and solving a linear system to obtain the locations of interparticle collisions. For inelastic systems, we show that the collision locations can always be uniquely determined. We also show that this is in sharp contrast to the case of elastic systems in which infinite families of self-similar orbits can coexist. © 2012 American Physical Society.