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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Volume 273, Pages 212–230 (Mi tm3286)  

This article is cited in 9 scientific papers (total in 9 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
Full-text PDF (239 kB) Citations (9)
References:
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.
Received in January 2010
English version:
Proceedings of the Steklov Institute of Mathematics, 2011, Volume 273, Pages 196–213
DOI: https://doi.org/10.1134/S0081543811040092
Bibliographic databases:
Document Type: Article
UDC: 517.984+531.36
Language: Russian
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
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\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}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2011
\vol 273
\pages 196--213
\crossref{https://doi.org/10.1134/S0081543811040092}
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  • This publication is cited in the following 9 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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