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

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:

UDC: 517.984+531.36

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, Trudy Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 212–230; Proc. Steklov Inst. Math., 273 (2011), 196–213

Citation in format AMSBIB
\Bibitem{Koz11} \by V.~V.~Kozlov \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 Trudy Mat. Inst. Steklova \yr 2011 \vol 273 \pages 212--230 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3286} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2893547} \zmath{https://zbmath.org/?q=an:1263.37059} \elib{https://elibrary.ru/item.asp?id=16456347} \transl \jour Proc. Steklov Inst. Math. \yr 2011 \vol 273 \pages 196--213 \crossref{https://doi.org/10.1134/S0081543811040092} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000295982500009} 

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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
2. Kozlov V.V., “Conservation Laws of Generalized Billiards That Are Polynomial in Momenta”, Russ. J. Math. Phys., 21:2 (2014), 226–241
3. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290
4. A. P. Markeev, “Ob ustoichivosti dvukhzvennoi traektorii paraboloidnogo bilyarda Birkgofa”, Nelineinaya dinam., 12:1 (2016), 75–90
5. A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 190–201
6. Treschev D., “A Locally Integrable Multi-Dimensional Billiard System”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5271–5284
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
8. Riviere A., Rouyer J., Vilcu C., Zamfirescu T., “Double Normals of Most Convex Bodies”, Adv. Math., 343 (2019), 245–272
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