Projective Dynamics and an Integrable Boltzmann Billiard Model

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.