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

2015/10/14 by Božidar Jovanović, Vladimir Jovanović, Jovanovic, Bozidar +1
Mathematics · Physics and Astronomy · #37J35 #37J55 #51M05 #70H06 #Advanced Differential Equations and Dynamical Systems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1510.04037

openalex publication_date 2015/10/14 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

The aim of the paper is to unify the efforts in the study of integrable\nbilliards within quadrics in flat and curved spaces and to explore further the\ninterplay of symplectic and contact integrability. As a starting point in this\ndirection, we consider virtual billiard dynamics within quadrics in\npseudo--Euclidean spaces. In contrast to the usual billiards, the incoming\nvelocity and the velocity after the billiard reflection can be at opposite\nsides of the tangent plane at the reflection point. In the symmetric case we\nprove noncommutative integrability of the system and give a geometrical\ninterpretation of integrals, an analog of the classical Chasles and Poncelet\ntheorems and we show that the virtual billiard dynamics provides a natural\nframework in the study of billiards within quadrics in projective spaces, in\nparticular of billiards within ellipsoids on the sphere mathbb S^n-1 and\nthe Lobachevsky space mathbb H^n-1.\n

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