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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2017 Volume 81, Issue 4, Pages 20–67 (Mi im8602)

This article is cited in 51 papers

Integrable topological billiards and equivalent dynamical systems

V. V. Vedyushkina (Fokicheva), A. T. Fomenko

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider several topological integrable billiards and prove that they are Liouville equivalent to many systems of rigid body dynamics. The proof uses the Fomenko–Zieschang theory of invariants of integrable systems. We study billiards bounded by arcs of confocal quadrics and their generalizations, generalized billiards, where the motion occurs on a locally planar surface obtained by gluing several planar domains isometrically along their boundaries, which are arcs of confocal quadrics. We describe two new classes of integrable billiards bounded by arcs of confocal quadrics, namely, non-compact billiards and generalized billiards obtained by gluing planar billiards along non-convex parts of their boundaries. We completely classify non-compact billiards bounded by arcs of confocal quadrics and study their topology using the Fomenko invariants that describe the bifurcations of singular leaves of the additional integral. We study the topology of isoenergy surfaces for some non-convex generalized billiards. It turns out that they possess exotic Liouville foliations: the integral trajectories of the billiard that lie on some singular leaves admit no continuous extension. Such billiards appear to be leafwise equivalent to billiards bounded by arcs of confocal quadrics in the Minkowski metric.

Keywords: integrable system, billiard, Liouville equivalence, Fomenko–Zieschang molecule.

UDC: 517.938.5

MSC: 37D50, 37J35, 70E40

Received: 15.09.2016

DOI: 10.4213/im8602


 English version:
Izvestiya: Mathematics, 2017, 81:4, 688–733

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