Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

Abstract

We study geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension $d$ and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is based on introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$ geometric types of a pencil into new relativistic quadrics of $d$ relativistic types. Deep insight of related geometry and combinatorics comes from our study of what we call discriminat sets of tropical lines ${\Sigma }^{+}$ and ${\Sigma }^{-}$ and their singularities. All of that enable usto get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.

29 pages, 7 figures

References in corpus (1)

• Geodesics on an ellipsoid in Minkowski space

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• On the circumcenters of triangular orbits in elliptic billards
• Pseudo-Euclidean Billiards within Confocal Curves on the Hyperboloid of One Sheet
• Heisenberg model in pseudo-Euclidean spaces II
• On 4-reflective complex analytic planar billiards
• Thesis manuscript: Projective and complex billiards, periodic orbits and Pfaffian systems