Integrability and chaos in the bouncing convex body model

We consider a dynamical system that models a strictly convex planar body with a smooth boundary, subjected to gravity and elastically reflected upon collision with a static wall. The system is equivalent to a billiard problem on a cylinder. It is shown that the system is integrable if and only if the body is a disk with its center of mass coincident with the geometric center. More specifically, for any other convex planar body with a smooth boundary, the existence of chaotic sets and Cantor circles on high-energy surfaces is proved. Additionally, the phenomenon of rigidity in more general rigid body impact systems is also discussed.