Abstract
We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.
Highlights
A mathematical billiard with moving boundary is a region of the plane instantaneously bounded by a closed curve changing with time
The billiard problem consists of the free motion of a point particle inside this region colliding elastically with the moving boundary
We show the nonexistence of some invariant curve using a criterion based on the variational approach of Aubry–Mather theory, in the spirit of what is done in [24]
Summary
A mathematical billiard with moving boundary is a region of the plane instantaneously bounded by a closed curve changing with time. A mathematical formulation of the problem was proposed by Ulam and is called the Fermi–Ulam model It describes the free motion of a particle between two parallel walls moving periodically. The dynamics of a time-dependent billiard whose boundary remains a convex curve can be described by a 4-dimensional exact symplectic map [15]. A considerable part of the paper is dedicated to get an explicit formulation of the generating function of the billiard map for large energies To this aim, we follow the idea, used in [16] for the (non-periodic) Fermi–Ulam model, that the generating function is given by the Lagrangian action of a solution of the Dirichlet problem between two consecutive impacts. The main results of Aubry–Mather theory used in the paper are collected in Appendix B
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