On the Hamiltonian property of linear dynamical systems in Hilbert space. (English. Russian original) Zbl 06769031

Treshchev, D. V.; Shkalikov, A. A.

Math. Notes 101, No. 6, 1033-1039 (2017); translation from Mat. Zametki 101, No. 6, 911-918 (2017).

Summary: 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.

MSC:

47-XX	Operator theory
70-XX	Mechanics of particles and systems
37-XX	Dynamical systems and ergodic theory

Keywords:

Hamiltonian system; Poisson bracket; symplectic structure

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