Authors: Dragović, Vladimir 
Radnović, Milena
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Geometry of integrable billiards and pencils of quadrics
Journal: Journal des Mathematiques Pures et Appliquees
Volume: 85
Issue: 6
First page: 758
Last page: 790
Issue Date: 1-Jan-2006
Rank: M21a
ISSN: 0021-7824
DOI: 10.1016/j.matpur.2005.12.002
Abstract: 
We study the deep interplay between geometry of quadrics in d-dimensional space and the dynamics of related integrable billiard systems. Various generalizations of Poncelet theorem are reviewed. The corresponding analytic conditions of Cayley's type are derived giving the full description of periodical billiard trajectories; among other cases, we consider billiards in arbitrary dimension d with the boundary consisting of arbitrary number k of confocal quadrics. Several important examples are presented in full details proving the effectiveness of the obtained results. We give a thorough analysis of classical ideas and results of Darboux and methodology of Lebesgue; we prove their natural generalizations, obtaining new interesting properties of pencils of quadrics. At the same time, we show essential connections between these classical ideas and the modern algebro-geometric approach in the integrable systems theory.
Keywords: Pencils of quadrics | Periodic billiard trajectories | Poncelet theorem
Publisher: Elsevier

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