Regular and Chaotic Dynamics

, Volume 23, Issue 1, pp 26–46 | Cite as

Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability



A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.


Hamiltonian system quadratic integrals integral cones degree of instability quantum systems Abelian integrals 

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  1. 1.
    Kozlov, V.V., Linear Systems with a Quadratic Integral, J. Appl. Math. Mech., 1992, vol. 56, no. 6, pp. 803–809; see also: Prikl. Mat. Mekh., 1992, vol. 56, no. 6, pp. 900–906.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Wimmer, H.K., Inertia Theorems for Matrices, Controllability, and Linear Vibrations, Linear Algebra and Appl., 1974, vol. 8, pp. 337–343.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Kozlov, V.V., On the Mechanism of Stability Loss, Differ. Equ., 2009, vol. 45, no. 4, pp. 510–519; see also: Differ. Uravn., 2009, vol. 45, no. 4, pp. 496–505.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Merkin, D.R., Gyroscopic Systems, 2nd rev. and enl. ed., Moscow: Nauka, 1974 (Russian).MATHGoogle Scholar
  5. 5.
    Kozlov, V.V., Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization, Funct. Anal. Appl., 2005, vol. 39, no. 4, pp. 271–283; see also: Funktsional. Anal. i Prilozhen., 2005, vol. 39, no. 4, pp. 32–47.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Williamson, J., On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems, Amer. J. Math., 1936, vol. 58, no. 1, pp. 141–163.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Williamson, J., An Algebraic Problem Involving the Involutory Integrals of Linear Dynamical Systems, Amer. J. Math., 1940, vol. 62, pp. 881–911.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Koçak, H., Linear Hamiltonian Systems Are Integrable with Quadratics, J. Math. Phys., 1982, vol. 23, no. 12, pp. 2375–2380.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.CrossRefMATHGoogle Scholar
  10. 10.
    Wintner, A., On the Linear Conservative Dynamical Systems, Ann. Mat. Pura Appl., 1934, vol. 13, no. 1, pp. 105–112.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Falconi, M., n − 1 Independent First Integrals for Linear Differential Systems, Qual. Theory Dyn. Syst., 2004, vol. 4, no. 2, pp. 233–254.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gorbuzov, V.N. and Pranevich, A. F., First Integrals of Ordinary Linear Differential Systems, arXiv:1201.4141 (2012).MATHGoogle Scholar
  13. 13.
    Berger, M., Geometry: In 2 Vols., Berlin: Springer, 1987.Google Scholar
  14. 14.
    Lakhadanov, V.M., On Stabilization of Potential Systems, J. Appl. Math. Mech., 1975, vol. 39, no. 1, pp. 45–50; see also: Prikl. Mat. Mekh., 1975, vol. 39, no. 1, pp. 53–58.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kozlov, V.V., Stabilization of the Unstable Equilibria of Charges by Intense Magnetic Fields, J. Appl. Math. Mech., 1997, vol. 61, no. 3, pp. 377–384; see also: Prikl. Mat. Mekh., 1997, vol. 61, no. 3, pp. 390–397.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bulatovič, R. M., The Stability of Linear Potential Gyroscopic Systems when the Potential Energy Has a Maximum, J. Appl. Math. Mech., 1997, vol. 61, no. 3, pp. 371–375; see also: Prikl. Mat. Mekh., 1997, vol. 61, no. 3, pp. 385–389.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Treshchev, D. V. and Shkalikov, A.A., On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space, Math. Notes, 2017, vol. 101, nos. 5–6, pp. 1033–1039; see also: Mat. Zametki, 2017, vol. 101, no. 6, pp. 911–918.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Faddeev, L.D. and Yakubovskii, O.A., Lectures on Quantum Mechanics for Mathematics Students, Stud. Math. Libr., vol. 47, Providence, R.I.: AMS, 2009.CrossRefMATHGoogle Scholar
  19. 19.
    Takhtajan, L.A., Quantum Mechanics for Matematicians, Grad. Stud. Math., vol. 95, Providence, R.I.: AMS, 2008.Google Scholar
  20. 20.
    Kozlov, V.V. and Treshchev, D. V., Polynomial Conservation Laws in Quantum Systems, Theoret. and Math. Phys., 2004, vol. 140, no. 3, pp. 1283–1298; see also: Teoret. Mat. Fiz., 2004, vol. 140, no. 3, pp. 460–479.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kozlov, V.V., Topological Obstructions to Existence of the Quantum Conservation Laws, Dokl. Math., 2005, vol. 71, no. 2, pp. 300–302; see also: Dokl. Akad. Nauk, 2005, vol 401, no. 5, pp. 603–606.MATHGoogle Scholar
  22. 22.
    Faddeev, L.D., What Is Complete Integrability in Quantum Mechanic, in Nonlinear Equations and Spectral Theory, M. S. Birman, N.N.Uraltseva (Eds.), Amer. Math. Soc. Transl. Ser. 2, vol. 220, Providence, R.I.: AMS, 2007, pp. 83–90.Google Scholar
  23. 23.
    Stöckmann, H.-J., Quantum Chaos: An Introduction, Cambridge: Cambridge Univ. Press, 1999.CrossRefMATHGoogle Scholar
  24. 24.
    Godunov, S.K., Modern Aspects of Linear Algebra, Providence, R.I.: AMS, 1998.MATHGoogle Scholar
  25. 25.
    Klein, F., Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Moscow: Nauka, 1989 (Russian).Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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