Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem

TL;DR

This paper develops L-A representations for geodesic and billiard flows on quadrics in pseudo-Euclidean spaces, providing a geometric interpretation of integrability similar to Chasles theorem and exploring virtual billiard reflections.

Contribution

It introduces L-A pairs for these flows in pseudo-Euclidean spaces and generalizes billiard problems to include virtual reflections, extending classical integrability results.

Findings

L-A representations for geodesic flows and billiards in pseudo-Euclidean spaces

Geometric interpretation of integrability akin to Chasles theorem

Generalization to virtual billiard reflections within arbitrary quadrics

Abstract

In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of the billiard within arbitrary quadric allowing virtual billiard reflections.