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Mathematical Notes

, Volume 101, Issue 5–6, pp 1033–1039 | Cite as

On the Hamiltonian property of linear dynamical systems in Hilbert space

  • D. V. Treshchev
  • A. A. Shkalikov
Volume 101, Number 6, June, 2017

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 ⊂ ker A*. For a bounded linear mapping x → Ωx possessing a first integral, sufficient conditions for the preservation of the (possibly degenerate) Poisson bracket are obtained.

Keywords

Hamiltonian system Poisson bracket symplectic structure 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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