Geometry of Integrable Billiards and Pencils of Quadrics
Abstract
We study the deep interplay between geometry of quadrics in ddimensional 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 demonstrating the effectiveness of the obtained results. We give a thorough analysis of classical ideas and results of Darboux and methodology of Lebesgue, and 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 algebrogeometric approach in the integrable systems theory.
 Publication:

arXiv eprints
 Pub Date:
 December 2005
 arXiv:
 arXiv:mathph/0512049
 Bibcode:
 2005math.ph..12049D
 Keywords:

 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Mathematical Physics;
 14H70;
 51N35;
 70H06
 EPrint:
 49 pages, 14 figures