D. V. Treschev, A. A. Shkalikov, “On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space”, Mat. Zametki, 101:6 (2017), 911–918; Math. Notes, 101:6 (2017), 1033

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Matematicheskie Zametki, 2017, Volume 101, Issue 6, Pages 911–918
DOI: https://doi.org/10.4213/mzm11520 (Mi mzm11520)

This article is cited in 12 scientific papers (total in 12 papers)

On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space

D. V. Treschev^a, A. A. Shkalikov^b

^a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^b Lomonosov Moscow State University

Full-text PDF (456 kB) Citations (12)

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DOI: https://doi.org/10.4213/mzm11520

Abstract: Conditions for the operator differential equation

$\dot x=Ax$ possessing a quadratic first integral

$(1/2)(Bx,x)$ to be Hamiltonian are obtained. In the finite-dimensional case, it suffices to require that

$\ker B \subset \ker A^*$ . For a bounded linear mapping

$x\to \Omega x$ possessing a first integral, sufficient conditions for the preservation of the (possibly degenerate) Poisson bracket are obtained.

Keywords: Hamiltonian system, Poisson bracket, symplectic structure.

Funding agency	Grant number
Russian Foundation for Basic Research	15-01-03747 16-01-00706
The work of the first author was supported by the Russian Foundation for Basic Research under grant 15-01-03747. The work of the second author was supported by the Russian Foundation for Basic Research under grant 16-01-00706.

Received: 07.09.2016

English version:
Mathematical Notes, 2017, Volume 101, Issue 6, Pages 1033–1039
DOI: https://doi.org/10.1134/S0001434617050303

Bibliographic databases:

Document Type: Article

UDC: 517.946

Language: Russian

Citation: D. V. Treschev, A. A. Shkalikov, “On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space”, Mat. Zametki, 101:6 (2017), 911–918; Math. Notes, 101:6 (2017), 1033–1039

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm11520

https://doi.org/10.4213/mzm11520

https://www.mathnet.ru/eng/mzm/v101/i6/p911

This publication is cited in the following 12 articles:

V. V. Kozlov, “Quantization of linear systems of differential equations with a quadratic invariant in a Hilbert space”, Russian Math. Surveys, 76:2 (2021), 357–359
V. V. Kozlov, “Symplectic geometry of the Koopman operator”, Dokl. Math., 104:1 (2021), 175–179
V. V. Kozlov, “Linear Nonautonomous Systems of Differential Equations With a Quadratic Integral”, Differ. Equ., 57:2 (2021), 173–181
V. V. Kozlov, “On the ergodic theory of equations of mathematical physics”, Russ. J. Math. Phys., 28:1 (2021), 73–83
V. V. Kozlov, “Linear system of differential equations with a quadratic invariant as the Schrödinger equation”, Dokl. Math., 103:1 (2021), 39–43
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, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140
V. E. Vladykina, A. A. Shkalikov, “Regular Ordinary Differential Operators with Involution”, Math. Notes, 106:5 (2019), 674–687
V. V. Kozlov, “Linear systems with quadratic integral and complete integrability of the Schrödinger equation”, Russian Math. Surveys, 74:5 (2019), 959–961
Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46
V. V. Kozlov, “Multi-Hamiltonian property of a linear system with quadratic invariant”, St. Petersburg Mathematical Journal, 30:5 (2019), 877–883

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

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