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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Volume 273, Pages 212–230
(Mi tm3286)
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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
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
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
Linking options:
https://www.mathnet.ru/eng/tm3286 https://www.mathnet.ru/eng/tm/v273/p212
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Abstract page: | 527 | Full-text PDF : | 88 | References: | 119 |
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