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 Theor. Appl. Mech., 2017, Volume 44, Issue 1, Pages 103–114 (Mi tam22)

Billiards on constant curvature spaces and generating functions for systems with constraints

Božidar Jovanović

Abstract: 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, generating functions, ellipsoidal billiards, Heisenberg spin model, skew-hodograph mapping.

 Funding Agency Grant Number Ministry of Education, Science and Technical Development of Serbia 174020 The research was supported by the Serbian Ministry of Science Project 174020, Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems.

DOI: https://doi.org/10.2298/TAM170523005J

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Bibliographic databases:

MSC: 37J10, 53D22, 51M05
Revised: 11.06.2017
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Citation: Božidar Jovanović, “Billiards on constant curvature spaces and generating functions for systems with constraints”, Theor. Appl. Mech., 44:1 (2017), 103–114

Citation in format AMSBIB
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