Time-dependent billiards

Abstract

This is an attempt to study mathematically billiards with moving boundaries. We assume that the boundary remains closed, regular and strictly convex, deforming periodically in time, in the normal direction. We describe the associated billiard diffeomorphism and the corresponding invariant measure. We discuss the stability of 2-periodic orbits and investigate the boundedness of the velocity in some precise examples. Finally, we present the Hamiltonian formalism and the symplectic structure, considering that a moving billiard is a billiard with rigid boundary on an augmented configuration space, with a singular metric.