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Izvestiya: Mathematics, 2017, Volume 81, Issue 4, Pages 688–733
DOI: https://doi.org/10.1070/IM8602
(Mi im8602)
 

This article is cited in 49 scientific papers (total in 49 papers)

Integrable topological billiards and equivalent dynamical systems

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

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
References:
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.
Received: 15.09.2016
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2017, Volume 81, Issue 4, Pages 20–67
DOI: https://doi.org/10.4213/im8602
Bibliographic databases:
Document Type: Article
UDC: 517.938.5
MSC: 37D50, 37J35, 70E40
Language: English
Original paper language: Russian
Citation: V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. RAN. Ser. Mat., 81:4 (2017), 20–67; Izv. Math., 81:4 (2017), 688–733
Citation in format AMSBIB
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  • https://doi.org/10.1070/IM8602
  • https://www.mathnet.ru/eng/im/v81/i4/p20
  • This publication is cited in the following 49 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:866
    Russian version PDF:172
    English version PDF:25
    References:72
    First page:52
     
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