Birkhoff and symplectic billiards are two classes of mathematical billiards, in which the billiard table is a strictly convex planar domain with smooth boundary. In the first case, the billiard ball reflects at the boundary in such a way that the angle of incidence is equal to the angle of reflection; in the second case, if x, y, z are three consecutive impact points on the boundary, the segment xy reflects to the segment yz if xz is parallel to the tangent line to the boundary at y. In this thesis we study the main properties of these mathematical billiards, pointing out similarities and differences. In particular, we show that in both cases the law describing the dynamics (the so called billiard map) admits a generating function and it is a monotone twist map. Then, we introduce the concepts of caustic (i.e., a curve in the billiard table with the property that if a trajectory of the billiard map is tangent to it, then it remains tangent after every each reflection) and of integrable billiards. Finally, we study Birkhoff and symplectic billiards in the framework of Aubry-Mather theory, which is concerned with the study of orbits of monotone twist maps minimizing the action functional.

Birkhoff and symplectic billiards; an overview

Testolina, Giorgia
2021/2022

Abstract

Birkhoff and symplectic billiards are two classes of mathematical billiards, in which the billiard table is a strictly convex planar domain with smooth boundary. In the first case, the billiard ball reflects at the boundary in such a way that the angle of incidence is equal to the angle of reflection; in the second case, if x, y, z are three consecutive impact points on the boundary, the segment xy reflects to the segment yz if xz is parallel to the tangent line to the boundary at y. In this thesis we study the main properties of these mathematical billiards, pointing out similarities and differences. In particular, we show that in both cases the law describing the dynamics (the so called billiard map) admits a generating function and it is a monotone twist map. Then, we introduce the concepts of caustic (i.e., a curve in the billiard table with the property that if a trajectory of the billiard map is tangent to it, then it remains tangent after every each reflection) and of integrable billiards. Finally, we study Birkhoff and symplectic billiards in the framework of Aubry-Mather theory, which is concerned with the study of orbits of monotone twist maps minimizing the action functional.
2021-02-26
89
billiards, Birkhoff, symplectic
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12608/23059