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Regul. Chaotic Dyn., 2018, Volume 23, Issue 1, Pages 26–46 (Mi rcd306)  

This article is cited in 13 scientific papers (total in 13 papers)

Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability

Valery V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: 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.

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

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations 01
This work was carried out within the framework of the scientific program of the Presidium of the Russian Academy of Sciences 01 “Fundamental Mathematics and its Applications”.


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MSC: 34A30
Received: 27.10.2017

Citation: Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46

Citation in format AMSBIB
\by Valery V. Kozlov
\paper Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 1
\pages 26--46

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    This publication is cited in the following articles:
    1. V. V. Kozlov, “Multi-Hamiltonian property of a linear system with quadratic invariant”, St. Petersburg Mathematical Journal, 30:5 (2019), 877–883  mathnet  crossref  mathscinet  isi  elib
    2. Zheglov A.B., Osipov D.V., “On First Integrals of Linear Hamiltonian Systems”, Dokl. Math., 98:3 (2018), 616–618  mathnet  crossref  mathscinet  zmath  isi  scopus
    3. V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. A. B. Zheglov, D. V. Osipov, “Lax pairs for linear Hamiltonian systems”, Siberian Math. J., 60:4 (2019), 592–604  mathnet  crossref  crossref  isi  elib
    5. V. V. Zharinov, “Hamiltonian operators with zero-divergence constraints”, Theoret. and Math. Phys., 200:1 (2019), 923–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. A. F. Pranevich, “On Poisson’s Theorem of Building First Integrals for Ordinary Differential Systems”, Rus. J. Nonlin. Dyn., 15:1 (2019), 87–96  mathnet  crossref  elib
    7. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness”, Regul. Chaotic Dyn., 24:3 (2019), 329–352  mathnet  crossref
    8. V. V. Kozlov, “Linear systems with quadratic integral and complete integrability of the Schrödinger equation”, Russian Math. Surveys, 74:5 (2019), 959–961  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. I. V. Volovich, “Complete integrability of quantum and classical dynamical systems”, P-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334  crossref  mathscinet  zmath  isi  scopus
    10. V. V. Kozlov, “First integrals and asymptotic trajectories”, Sb. Math., 211:1 (2020), 29–54  mathnet  crossref  crossref  mathscinet  isi  elib
    11. V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics”, Russian Math. Surveys, 75:3 (2020), 445–494  mathnet  crossref  crossref  mathscinet  isi  elib
    12. V. V. Kozlov, “The Liouville Equation as a Hamiltonian System”, Math. Notes, 108:3 (2020), 339–343  mathnet  crossref  crossref  isi  elib
    13. V. V. Kozlov, “The stability of circulatory systems”, Dokl. Phys., 65:9 (2020), 323–325  crossref  isi  scopus
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