Authors: Jovanović, Božidar 
Jovanović, Vladimir
Title: Virtual billiards in pseudo Euclidean spaces: Discrete hamiltonian and contact integrability
Journal: Discrete and Continuous Dynamical Systems- Series A
Volume: 37
Issue: 10
First page: 5163
Last page: 5190
Issue Date: 1-Oct-2017
Rank: M21
ISSN: 1078-0947
DOI: 10.3934/dcds.2017224
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 Sn-1 and the Lobachevsky space ℍn-1.
Keywords: Billiards | Confocal quadrics | Discrete integrability | Lax representation | Poncelet theorem | Virtual reflection
Publisher: American Institute of Mathematical Sciences

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