Authors: Jovanović, Božidar 
Title: Billiards on constant curvature spaces and generating functions for systems with constraints
Journal: Theoretical and Applied Mechanics
Volume: 44
Issue: 1
First page: 103
Last page: 114
Issue Date: 1-Jan-2017
Rank: M24
ISSN: 1450-5584
DOI: 10.2298/TAM170523005J
In this note we consider a method of generating functions for systems with constraints and, as an example, we prove that the billiard mappings for billiards on the Euclidean space, sphere, and the Lobachevsky space are sympletic. Further, by taking a quadratic generating function we get the skew-hodograph mapping introduced by Moser and Veselov, which relates the ellipsoidal billiards in the Euclidean space with the Heisenberg magnetic spin chain model on a sphere. We define analogous mapping for the ellipsoidal billiard on the Lobachevsky space. It relates the billiard with the Heisenberg spin model on the light-like cone in the Lorentz-Poincare-Minkowski space.
Keywords: Dirac brackets | Ellipsoidal billiards | Generating functions | Heisenberg spin model | Skew-hodograph mapping
Publisher: Serbian Society for Mechanics
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 

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