math.DS — Dynamical Systems

Collision Systems, with Applications to Geodesic Networks and Point Billiards

Contributed by Alec Shelley, Henry Bosch

A collision system is a finite set of point particles with masses, moving freely with velocities given by the gradient of a convex energy function of the momentum, whose collisions conserve total momentum and total energy and are otherwise unconstrained. Stationary geodesic networks near a line of multiplicity $n$, point billiards with momentum- and energy-conserving collisions, and relativistic point particles are collision systems. We prove that the number of collisions of a collision system is at most exponential in its total mass, with an explicit base, whenever the energy is close enough to the Newtonian one, and that it can be exponential for every energy on a line. For geodesic networks this replaces the previously reported bounds, of order $n^2$ from below and $n^{n^2}$ from above, by exponential bounds on both sides; for point billiards it gives explicit exponential bounds. The exponential growth rates remain open.