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Mat. Zametki, 2017, Volume 101, Issue 6, Pages 911–918 (Mi mz11520)  

This article is cited in 3 scientific papers (total in 3 papers)

On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space

D. V. Trescheva, A. A. Shkalikovb

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

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.


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

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English version:
Mathematical Notes, 2017, 101:6, 1033–1039

Bibliographic databases:

Document Type: Article
UDC: 517.946
Received: 07.09.2016

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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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Kozlov, “Linear Hamiltonian systems: quadratic integrals, singular subspaces and stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46  mathnet  crossref  mathscinet
    2. V. V. Kozlov, “Multigamiltonovost lineinoi sistemy s kvadratichnym invariantom”, Algebra i analiz, 30:5 (2018), 159–168  mathnet
    3. V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140  mathnet  crossref  crossref  adsnasa  isi  elib
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