Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard

  • V. V. Kozlov


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.


Quadratic Form Variational Principle STEKLOV Institute Symmetric Operator Length Function 
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  1. 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
  2. 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).zbMATHGoogle Scholar
  3. 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)].zbMATHMathSciNetGoogle Scholar
  4. 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. 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
  6. 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)].MathSciNetGoogle Scholar
  7. 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)].MathSciNetGoogle Scholar
  8. 8.
    F. R. Gantmakher, The Theory of Matrices (Nauka, Moscow, 1988; Am. Math. Soc., Providence, RI, 2000).Google Scholar
  9. 9.
    R. S. Mackay and J. D. Meiss, “Linear Stability of Periodic Orbits in Lagrangian Systems,” Phys. Lett. A 98(3), 92–94 (1983).CrossRefMathSciNetGoogle Scholar
  10. 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
  11. 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).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Poincaré, “Sur les déterminants d’ordre infini,” Bull. Soc. Math. France 14, 77–90 (1886).MathSciNetGoogle Scholar
  13. 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).zbMATHGoogle Scholar
  14. 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)].MathSciNetGoogle Scholar
  15. 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)].MathSciNetGoogle Scholar
  16. 16.
    L. S. Pontryagin, “Hermitian Operators in a Space with Indefinite Metric,” Izv. Akad. Nauk SSSR, Ser. Mat. 8(6), 243–280 (1944).zbMATHGoogle Scholar
  17. 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)].zbMATHMathSciNetGoogle Scholar
  18. 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)].MathSciNetGoogle Scholar
  19. 19.
    H. K. Wimmer, “Inertia Theorems for Matrices, Controllability, and Linear Vibrations,” Linear Algebra Appl. 8, 337–343 (1974).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 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. 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

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© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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