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Tr. Mat. Inst. Steklova, 2011, Volume 273, Pages 212–230 (Mi tm3286)  

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

Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard

V. V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

Abstract: A linearized problem of stability of simple periodic motions with elastic reflections is considered: a particle moves along a straight-line segment that is orthogonal to the boundary of a billiard at its endpoints. In this problem issues from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics, etc.) are naturally intertwined. Multidimensional variants of Hill's formula, which relates the dynamic and geometric properties of a periodic trajectory, are discussed. Stability conditions are expressed in terms of the geometric properties of the boundary of a billiard. In particular, it turns out that a nondegenerate two-link trajectory of maximum length is always unstable. The degree of instability (the number of multipliers outside the unit disk) is estimated. The estimates are expressed in terms of the geometry of the caustic and the Morse indices of the length function of this trajectory.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 273, 196–213

Bibliographic databases:

Document Type: Article
UDC: 517.984+531.36
Received in January 2010

Citation: V. V. Kozlov, “Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard”, Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Tr. Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 212–230; Proc. Steklov Inst. Math., 273 (2011), 196–213

Citation in format AMSBIB
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\paper Problem of stability of two-link trajectories in a~multidimensional Birkhoff billiard
\inbook Modern problems of mathematics
\bookinfo Collected papers. In honor of the 75th anniversary of the Institute
\serial Tr. Mat. Inst. Steklova
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\vol 273
\pages 212--230
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
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\pages 196--213
\crossref{https://doi.org/10.1134/S0081543811040092}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. N. Davletshin, “Hill’s formula for $g$-periodic trajectories of Lagrangian systems”, Trans. Moscow Math. Soc., 74 (2013), 65–96  mathnet  crossref  mathscinet  zmath  elib
    2. Kozlov V.V., “Conservation Laws of Generalized Billiards That Are Polynomial in Momenta”, Russ. J. Math. Phys., 21:2 (2014), 226–241  crossref  mathscinet  zmath  isi  elib
    3. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. P. Markeev, “Ob ustoichivosti dvukhzvennoi traektorii paraboloidnogo bilyarda Birkgofa”, Nelineinaya dinam., 12:1 (2016), 75–90  mathnet
    5. A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 190–201  mathnet  crossref  crossref  mathscinet  isi  elib
    6. Treschev D., “A Locally Integrable Multi-Dimensional Billiard System”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5271–5284  crossref  mathscinet  zmath  isi
    7. A. N. Kirillov, R. V. Alkin, “Ustoichivost periodicheskikh bilyardnykh traektorii v treugolnike”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 18:1 (2018), 25–39  mathnet  crossref  elib
    8. Riviere A., Rouyer J., Vilcu C., Zamfirescu T., “Double Normals of Most Convex Bodies”, Adv. Math., 343 (2019), 245–272  crossref  mathscinet  zmath  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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