Projective Dynamics and an Integrable Boltzmann Billiard Model

TL;DR

This paper explains the integrability of a Boltzmann billiard model using projective dynamics, revealing connections to energy conservation and extending to other Kepler-Coulomb related billiard systems on spheres.

Contribution

It introduces a novel viewpoint of projective dynamics to establish integrability and extends the class of integrable billiard models related to Kepler-Coulomb problems.

Findings

The integrability of the Boltzmann billiard model is explained via projective dynamics.

A new family of integrable billiard models on the sphere is identified.

The approach links the first integral to the energy of a system on a hemisphere.

Abstract

The aim of this note is to explain the integrability of an integrable Boltzmann billiard model, previously established by Gallavotti and Jauslin in arXiv:2008.01955, alternatively via the viewpoint of projective dynamics. The additional first integral is shown to be related to the energy of a corresponding system on a hemisphere. We show that this viewpoint applies to certain other billiard models in the plane defined through Kepler-Coulomb problems as well. The approach also leads to a family of integrable billiard models on the sphere defined through the spherical Kepler-Coulomb problem.