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Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard. (English. Russian original) Zbl 1263.37059
Proc. Steklov Inst. Math. 273, 196-213 (2011); translation from Tr. Mat. Inst. Steklova 273, 212-230 (2011).
The author studies the stability of periodic 2-orbits of the billiard dynamics on a higher dimensional convex table \(\mathcal{D}\) and provides several viewpoints from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics). Note that the caustics of a billiard table \(\mathcal{D}\) consists of the focal points of \(\Gamma=\partial\mathcal{D}\) with respect to points \(z\in\Gamma\).
Moreover, the author gives an estimate of the dimension of the unstable subspace at these orbits in terms of the geometry of the caustic and the Morse indices of the length function. More precisely, let \(z\) be a point on the boundary \(\Gamma\) and \(I(z)\) be the segment on \(\mathcal{D}\) intersecting \(\Gamma\) orthogonally at \(z\). Denote by \(L(z)\) the length of the segment \(I(z)\). Then Theorem 1 states that, under some mild conditions, the dimension of the unstable subspace at a periodic 2-orbit equals to the Morse index.
Theorem 2 calculates the characteristic polynomial of the Poincaré operator as the determinant of a matrix of only half the dimension. Hill’s formula is also generalized to the higher dimensional case. Sufficient conditions are given to ensure that all multipliers lie on the unit circle.

MSC:
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
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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). · Zbl 0255.35002
[2] V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Mosk. Gos. Univ., Moscow, 1991; Am. Math. Soc., Providence, RI, 1991). · Zbl 0751.70009
[3] V. V. Kozlov, ”Two-Link Billiard Trajectories: Extremal Properties and Stability,” Prikl. Mat. Mekh. 64(6), 942–946 (2000) [J. Appl. Math. Mech. 64, 903–907 (2000)]. · Zbl 0979.37011
[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. · Zbl 0853.58071
[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)].
[6] V. V. Kozlov and I. I. Chigur, ”The Stability of Periodic Trajectories of a Billiard Ball in Three Dimensions,” Prikl. Mat. Mekh. 55(5), 713–717 (1991) [J. Appl. Math. Mech. 55, 576–580 (1991)]. · Zbl 0788.70011
[7] V. V. Kozlov, ”A Constructive Method of Establishing the Validity of the Theory of Systems with Non-retaining Constraints,” Prikl. Mat. Mekh. 52(6), 883–894 (1988) [J. Appl. Math. Mech. 52, 691–699 (1988)]. · Zbl 0711.70021
[8] F. R. Gantmakher, The Theory of Matrices (Nauka, Moscow, 1988; Am. Math. Soc., Providence, RI, 2000). · Zbl 0666.15002
[9] R. S. Mackay and J. D. Meiss, ”Linear Stability of Periodic Orbits in Lagrangian Systems,” Phys. Lett. A 98(3), 92–94 (1983).
[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)]. · Zbl 0667.58033
[11] G. W. Hill, ”On the Part of the Motion of the Lunar Perigel Which Is a Function of the Mean Motion of the Sun and Moon,” Acta Math. 8, 1–36 (1886). · JFM 18.1106.01
[12] H. Poincaré, ”Sur les déterminants d’ordre infini,” Bull. Soc. Math. France 14, 77–90 (1886). · JFM 18.0117.01
[13] H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Librairie Scientifique et Technique Albert Blanchard, Paris, 1987), Vol. 3; Engl. transl.: H. Poincaré, New Methods in Celestial Mechanics, Vol. 3 (Am. Inst. Phys., Bristol, 1993).
[14] V. V. Kozlov, ”Remarks on Eigenvalues of Real Matrices,” Dokl. Akad. Nauk 403(5), 589–592 (2005) [Dokl. Math. 72 (1), 567–569 (2005)].
[15] V. V. Kozlov and A. A. Karapetyan, ”On the Stability Degree,” Diff. Uravn. 41(2), 186–192 (2005) [Diff. Eqns. 41, 195–201 (2005)]. · Zbl 1090.34564
[16] L. S. Pontryagin, ”Hermitian Operators in a Space with Indefinite Metric,” Izv. Akad. Nauk SSSR, Ser. Mat. 8(6), 243–280 (1944). · Zbl 0061.26004
[17] M. G. Krein, ”On an Application of the Fixed-Point Principle in the Theory of Linear Transformations of Spaces with an Indefinite Metric,” Usp. Mat. Nauk 5(2), 180–190 (1950) [Am. Math. Soc. Transl., Ser. 2, 1, 27–35 (1955)].
[18] V. I. Arnol’d, ”Conditions for Nonlinear Stability of Stationary Plane Curvilinear Flows of an Ideal Fluid,” Dokl. Akad. Nauk SSSR 162(5), 975–978 (1965) [Sov. Math., Dokl. 6, 773–777 (1965)].
[19] H. K. Wimmer, ”Inertia Theorems for Matrices, Controllability, and Linear Vibrations,” Linear Algebra Appl. 8, 337–343 (1974). · Zbl 0288.15015
[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. · Zbl 0860.47009
[21] V. V. Kozlov, ”On the Mechanism of Stability Loss,” Diff. Uravn. 45(4), 496–505 (2009) [Diff. Eqns. 45, 510–519 (2009)].
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