A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics

Abstract

A new class of integrable billiard systems, called generalized billiards, is discovered. These are billiards in domains formed by gluing classical billiard domains along pieces of their boundaries. (A classical billiard domain is a part of the plane bounded by arcs of confocal quadrics.) On the basis of the Fomenko-Zieschang theory of invariants of integrable systems, a full topological classification of generalized billiards is obtained, up to Liouville equivalence.Bibliography: 18 titles.

Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

“…Since these two integrals are functionally independent and are in involution with respect to the standard Poisson bracket, we can state that the topological billiard system is piecewise integrable in the sense of Liouville. For more detailed definitions, see V. V. Fokicheva's work [30].…”

“…Fokicheva classified all topological billiards bounded by the arcs of confocal conics (the families of confocal ellipses and hyperbolas and the confocal parabolas) [29,30]. Further, Fokicheva investigated the topology of Liouville foliations on isoenergy surfaces for such billiards by calculating the Fomenko-Zieschang invariants of these systems.…”

Liouville Foliation of Topological Billiards in the Minkowski Plane

“…Dragović and M. Radnović [3,4] and V. V. Fokicheva gave a full Liouville classification of flat billiards bounded by arcs of confocal quadrics. Then V. V. Fokicheva [6] considered a topological billiard glued from boundaries that are planar along arcs.…”

“…Since these integrals are functionally independent and are in involution with respect to the standard Poisson bracket, we consider the system of a topological billiard to be piecewise Liouville integrable. For detailed definitions we refer the reader to [6].…”

Billiards and integrable systems

“…Vedyushkina succeeded in enlarging the class of billiards under investigation in a crucial was by allowing piecewise planar two-dimensional tables glued of planar table along common boundary arcs. The classes of topological (generalized) billiards [31] and billiard books [32] were introduced. Topological billiards are homeomorphic to orientable manifolds (although they are only piecewise planar).…”