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

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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### References:

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