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

This article is cited in 18 scientific papers (total in 18 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.

DOI: https://doi.org/10.4213/im8602

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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

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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathscinet  zmath  isi  elib  scopus
    2. V. A. Trifonova, “Partially symmetric height atoms”, Moscow University Mathematics Bulletin, 73:2 (2018), 71–78  mathnet  crossref  mathscinet  zmath  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 University Mathematics Bulletin, 73:3 (2018), 103–110  mathnet  crossref  mathscinet  zmath  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  adsnasa  isi  elib
    7. V. A. Kibkalo, “Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra $\operatorname{so}(4)$”, Sb. Math., 210:5 (2019), 625–662  mathnet  crossref  crossref  adsnasa  isi  elib
    8. A. T. Fomenko, V. V. Vedyushkina, “Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: recent results”, Theor. Appl. Mech., 46:1 (2019), 47–63  mathnet  crossref
    9. V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173  mathnet  crossref  crossref  adsnasa  isi  elib
    10. V. V. Vedyushkina, A. T. Fomenko, “Topological obstacles to the realizability of integrable Hamiltonian systems by billiards”, Dokl. Math., 100:2 (2019), 463–466  crossref  zmath  isi  scopus
    11. E. E. Karginova, “Sloenie Liuvillya topologicheskikh billiardov na ploskosti Minkovskogo”, Fundament. i prikl. matem., 22:6 (2019), 123–150  mathnet
    12. S. E. Pustovoitov, “Topologicheskii analiz billiarda v ellipticheskom koltse v potentsialnom pole”, Fundament. i prikl. matem., 22:6 (2019), 201–225  mathnet
    13. E. E. Karginova, “Billiards bounded by arcs of confocal quadrics on the Minkowski plane”, Sb. Math., 211:1 (2020), 1–28  mathnet  crossref  crossref  isi
    14. V. V. Vedyushkina, “Integrable billiard systems realize toric foliations on lens spaces and the 3-torus”, Sb. Math., 211:2 (2020), 201–225  mathnet  crossref  crossref  isi  elib
    15. I. F. Kobtsev, “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Sb. Math., 211:7 (2020), 987–1013  mathnet  crossref  crossref  isi
    16. S. S. Nikolaenko, “Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds”, Sb. Math., 211:8 (2020), 1127–1158  mathnet  crossref  crossref  isi
    17. V. A. Moskvin, “Algoritmicheskoe postroenie dvumernykh osobykh sloev atomov bilyardov v nevypuklykh oblastyakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 3, 3–12  mathnet
    18. G. V. Belozerov, “Topologicheskaya klassifikatsiya integriruemykh geodezicheskikh billiardov na kvadrikakh v trekhmernom evklidovom prostranstve”, Matem. sb., 211:11 (2020), 3–40  mathnet  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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