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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 4, Pages 20–67 (Mi izv8602)  

This article is cited in 8 scientific papers (total in 8 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.


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

Bibliographic databases:

UDC: 517.938.5
MSC: 37D50, 37J35, 70E40
Received: 15.09.2016

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
\by V.~V.~Vedyushkina (Fokicheva), A.~T.~Fomenko
\paper Integrable topological billiards and equivalent dynamical systems
\jour Izv. RAN. Ser. Mat.
\yr 2017
\vol 81
\issue 4
\pages 20--67
\jour Izv. Math.
\yr 2017
\vol 81
\issue 4
\pages 688--733

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    This publication is cited in the following articles:
    1. V. V. Vedyushkina, “The Liouville foliation of nonconvex topological billiards”, Dokl. Math., 97:1 (2018), 1–5  mathnet  crossref  crossref  zmath  isi  elib  scopus
    2. V. A. Trifonova, “Partially symmetric height atoms”, Moscow Univ. Math. Bull., 73:2 (2018), 71–78  mathnet  crossref  isi
    3. V. V. Vedyushkina, A. T. Fomenko, I. S. Kharcheva, “Modeling nondegenerate bifurcations of closures of solutions for integrable systems with two degrees of freedom by integrable topological billiards”, Dokl. Math., 97:2 (2018), 174–176  mathnet  crossref  crossref  zmath  isi  elib  scopus
    4. V. A. Moskvin, “Topology of Liouville bundles of integrable billiard in non-convex domains”, Moscow Univ. Math. Bull., 73:3 (2018), 103–110  mathnet  crossref  isi
    5. V. V. Vedyushkina, I. S. Kharcheva, “Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems”, Sb. Math., 209:12 (2018), 1690–1727  mathnet  crossref  crossref  adsnasa  isi  elib
    6. V. V. Vedyushkina, “The Fomenko–Zieschang invariants of nonconvex topological billiards”, Sb. Math., 210:3 (2019), 310–363  mathnet  crossref  crossref  elib
    7. V. A. Kibkalo, “Topologicheskaya klassifikatsiya sloenii Liuvillya dlya integriruemogo sluchaya Kovalevskoi na algebre Li $\operatorname{so}(4)$”, Matem. sb., 210:5 (2019), 3–40  mathnet  crossref  elib
    8. Fomenko A.T. Vedyushkina V.V., “Singularities of Integrable Liouville Systems, Reduction of Integrals to Lower Degree and Topological Billiards: Recent Results”, Theor. Appl. Mech., 46:1 (2019), 47–63  crossref  isi
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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