Billiards in Finsler and Minkowski geometries

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• 作者：Gutkin ; Eugene ; Tabachnikov ; Serge
• 关键词：70F35 ; 53B40 ; Dynamical systems ; Finsler and Minkowski geometries ; Billiard reflection ; Legendre transform ; Symplectic structure ; Generating function ; Mean free path ; Mirror equation ; Caustic
• 刊名：Journal of Geometry and Physics
• 出版年：2002
• 出版时间：January, 2002
• 年：2002
• 卷：40
• 期：3-4
• 页码：277-301
• 全文大小：181 K

文摘

We begin the study of billiard dynamics in Finsler geometry. We deduce the Finsler billiard reflection law from the “least action principle”, and extend the basic properties of Riemannian and Euclidean billiards to the Finsler and Minkowski settings, respectively. We prove that the Finsler billiard map is a symplectomorphism, and compute the mean free path of the Finsler billiard ball. For the planar Minkowski billiard we obtain the mirror equation, and extend the Mather’s non-existence of caustics result. We establish an orbit-to-orbit duality for Minkowski billiards.