Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

2011/08/31 by Vladimir Dragovic, Milena Radnovic · 1 citation
Mathematics · Physics and Astronomy · #math.AG #math-ph #math.MP

paper · pdf · doi:10.1016/j.aim.2012.06.004

published as Advances in Mathematics 231 (2012), 1173-1201 · 29 pages, 7 figures

arxiv created 2012/05/11 · arxiv updated 2013/01/01

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 Σ⁺ and Σ^- 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.

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