Geometry of Integrable Billiards and Pencils of Quadrics

TL;DR

This paper explores the geometric and algebraic structures underlying integrable billiard systems in multiple dimensions, extending classical theorems and deriving new conditions for periodic trajectories in complex quadrics.

Contribution

It provides new analytic conditions for periodic billiard trajectories in arbitrary dimensions and generalizes classical geometric theorems using modern algebraic methods.

Findings

Derived Cayley's type conditions for billiard trajectories

Extended Poncelet theorem to higher dimensions and multiple confocal quadrics

Connected classical geometry with modern integrable systems theory

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 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 algebro-geometric approach in the integrable systems theory.