On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies

Details On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies

2002-10-10

arxiv.org/abs/math-ph/0210019v1

Journal of Geometry and Physics 47 (2003) 221-234

Physics

Mathematical Physics

14 pages, to appear in Journal of Geometry and Physics

Scientific paper

10.1016/S0393-0440(02)00219-X

We derive Cayley's type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We explain this coincidence by using theory of geodesically equivalent metrics and show that Lobachevsky and Euclidean elliptic billiards can be naturally considered as a part of a hierarchy of integrable elliptical billiards.

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