Authors: Dragović, Vladimir 
Radnović, Milena
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics
Journal: Advances in Mathematics
Volume: 231
Issue: 3-4
First page: 1173
Last page: 1201
Issue Date: 1-Oct-2012
Rank: M21
ISSN: 0001-8708
DOI: 10.1016/j.aim.2012.06.004
We study the geometry of confocal quadrics in pseudo-Euclidean spaces of 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 the introduction of a new discrete combinatorial geometric structure associated with a confocal pencil of quadrics, a colouring in d colours. This is used to decompose quadrics of d+1 geometric types of a pencil into new relativistic quadrics of d relativistic types. A study of what we term discriminant sets of tropical lines σ + and σ - and their singularities provides insight into the related geometry and combinatorics. This yields an analytic criterion describing all periodic billiard trajectories, including light-like trajectories as a case of special interest.
Keywords: Confocal quadrics | Light-like billiard trajectories | Minkowski space | Periodic billiard trajectories | Poncelet theorem | Tropic curves
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Mathematical Physics Group of the University of Lisbon (Project Probabilistic Approach to Finite- and Infinite-Dimensional Dynamical Systems, PTDC/MAT/104173/2008)

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