
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 finitedimensional quantum systems.
Keywords:
Hamiltonian system, quadratic integrals, integral cones, degree of instability, quantum systems, Abelian integrals
DOI:
https://doi.org/10.1134/S1560354718010033
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MSC: 34A30 Received: 27.10.2017 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
Citation in format AMSBIB
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\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 2646
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V. V. Kozlov, “MultiHamiltonian property of a linear system with quadratic invariant”, St. Petersburg Mathematical Journal, 30:5 (2019), 877–883

Zheglov A.B., Osipov D.V., “On First Integrals of Linear Hamiltonian Systems”, Dokl. Math., 98:3 (2018), 616–618

V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140

A. B. Zheglov, D. V. Osipov, “Lax pairs for linear Hamiltonian systems”, Siberian Math. J., 60:4 (2019), 592–604

V. V. Zharinov, “Hamiltonian operators with zerodivergence constraints”, Theoret. and Math. Phys., 200:1 (2019), 923–937

A. F. Pranevich, “On Poisson’s Theorem of Building First Integrals for Ordinary Differential Systems”, Rus. J. Nonlin. Dyn., 15:1 (2019), 87–96

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

V. V. Kozlov, “Linear systems with quadratic integral and complete integrability of the Schrödinger equation”, Russian Math. Surveys, 74:5 (2019), 959–961

I. V. Volovich, “Complete integrability of quantum and classical dynamical systems”, PAdic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334

V. V. Kozlov, “First integrals and asymptotic trajectories”, Sb. Math., 211:1 (2020), 29–54

V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics”, Russian Math. Surveys, 75:3 (2020), 445–494

V. V. Kozlov, “The Liouville Equation as a Hamiltonian System”, Math. Notes, 108:3 (2020), 339–343

V. V. Kozlov, “The stability of circulatory systems”, Dokl. Phys., 65:9 (2020), 323–325

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