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

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”.

DOI: https://doi.org/10.1134/S1560354718010033

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MSC: 34A30
Accepted:30.11.2017
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Citation: Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Zheglov A.B., Osipov D.V., “On First Integrals of Linear Hamiltonian Systems”, Dokl. Math., 98:3 (2018), 616–618
3. V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140
4. A. B. Zheglov, D. V. Osipov, “Lax pairs for linear Hamiltonian systems”, Siberian Math. J., 60:4 (2019), 592–604
5. V. V. Zharinov, “Hamiltonian operators with zero-divergence constraints”, Theoret. and Math. Phys., 200:1 (2019), 923–937
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
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
8. V. V. Kozlov, “Linear systems with quadratic integral and complete integrability of the Schrödinger equation”, Russian Math. Surveys, 74:5 (2019), 959–961
9. I. V. Volovich, “Complete integrability of quantum and classical dynamical systems”, P-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334
10. V. V. Kozlov, “First integrals and asymptotic trajectories”, Sb. Math., 211:1 (2020), 29–54
11. V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics”, Russian Math. Surveys, 75:3 (2020), 445–494
12. V. V. Kozlov, “The Liouville Equation as a Hamiltonian System”, Math. Notes, 108:3 (2020), 339–343
13. V. V. Kozlov, “The stability of circulatory systems”, Dokl. Phys., 65:9 (2020), 323–325
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