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)

Show full item record


checked on Jun 23, 2024

Page view(s)

checked on May 9, 2024

Google ScholarTM




Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.