On elliptical billiards in the Lobachevsky space and associated geodesic hierarchies

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• 作者：Dragovic ; Vladimir ; Jovanovic ; Bozidar ; Radnovic ; Milena
• 关键词：70H06 ; 53D25 ; Geometry of integrable systems ; Dynamical systems ; Integrable billiards ; Poncelet’ ; s theorem ; Cayley’ ; s condition ; Spectral curve ; Geodesic hierarchy ; Separable perturbation
• 刊名：Journal of Geometry and Physics
• 出版年：2003
• 期刊代码：29_03930440
• 类别：ph
• 出版时间：August, 2003
• 卷：47
• 期：2-3
• 页码：221-234
• 文件大小：133 K

摘要

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.