Geodesic
On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space
Matematičeskie zametki, Tome 101 (2017) no. 6, pp. 911-918

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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, symplectic structure.
Mots-clés : Poisson bracket 
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     author = {D. V. Treschev and A. A. Shkalikov},
     title = {On the {Hamiltonian} {Property} of {Linear} {Dynamical} {Systems} in {Hilbert} {Space}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {911--918},
     publisher = {mathdoc},
     volume = {101},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_101_6_a10/}
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D. V. Treschev; A. A. Shkalikov. On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space. Matematičeskie zametki, Tome 101 (2017) no. 6, pp. 911-918. http://geodesic.mathdoc.fr/item/MZM_2017_101_6_a10/