Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard
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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.
KeywordsQuadratic Form Variational Principle STEKLOV Institute Symmetric Operator Length Function
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- 1.V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory (Nauka, Moscow, 1972; Alpha Sci. Int., Oxford, 2008).Google Scholar
- 4.A. A. Markeev, “The Method of Pointwise Mappings in the Stability Problem of Two-Segment Trajectories of the Birkhoff Billiards,” in Dynamical Systems in Classical Mechanics (Am. Math. Soc., Providence, RI, 1995), AMS Transl., Ser. 2, 168, pp. 211–226.Google Scholar
- 5.A. P. Markeev, “Area-Preserving Mappings and Their Applications to the Dynamics of Systems with Collisions,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 37–54 (1996) [Mech. Solids 31 (2), 32–47 (1996)].Google Scholar
- 8.F. R. Gantmakher, The Theory of Matrices (Nauka, Moscow, 1988; Am. Math. Soc., Providence, RI, 2000).Google Scholar
- 10.D. V. Treshchev, “On the Question of the Stability of the Periodic Trajectories of Birkhoff’s Billiard,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 2, 44–50 (1988) [Moscow Univ. Mech. Bull. 43 (2), 28–36 (1988)].Google Scholar
- 20.A. A. Shkalikov, “Operator Pencils Arising in Elasticity and Hydrodynamics: The Instability Index Formula,” in Recent Developments in Operator Theory and Its Applications (Birkhäuser, Basel, 1996), Oper. Theory, Adv. Appl. 87, pp. 358–385.Google Scholar
- 21.V. V. Kozlov, “On the Mechanism of Stability Loss,” Diff. Uravn. 45(4), 496–505 (2009) [Diff. Eqns. 45, 510–519 (2009)].Google Scholar