On differential equations of integrable billiard tables

TL;DR

This paper develops a method to derive differential equations characterizing billiard tables that support integrable billiard systems, with applications to wire billiards and specific surfaces in three-dimensional space.

Contribution

It introduces a novel approach to find differential equations for tables ensuring integrability of billiard systems, applicable to various geometries.

Findings

Derived differential equations for integrable billiard tables.

Identified conditions for integrability in wire billiards.

Constructed examples of integrable surfaces in ${ m R}^3$.

Abstract

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ${\mathbb R}^3$ with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ${\mathbb R}^3$ homeomorphic to a torus, being a table of an integrable billiard.