Virtual billiards in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability

TL;DR

This paper investigates virtual billiard dynamics in pseudo-Euclidean spaces, establishing integrability and geometric properties, and connecting these findings to classical billiard theorems and billiards in curved spaces.

Contribution

It introduces a unified framework for virtual billiards in pseudo-Euclidean spaces, proving noncommutative integrability and providing geometric interpretations of integrals.

Findings

Proves noncommutative integrability of virtual billiards in symmetric cases.

Provides geometric interpretation of integrals analogous to classical theorems.

Shows the relevance of virtual billiards to billiards in projective and curved spaces.

Abstract

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo--Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere $\mathbb S^{n-1}$ and the Lobachevsky space $\mathbb H^{n-1}$.