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Abstract <p>We study Hamiltonian flows in a real separable Hilbert spaceendowed with a symplectic structure. Measures on the Hilbert spacethat are invariant with respect to the flows of completelyintegrable Hamiltonian systems are investigated. Invariantmeasures are used to study linear Hamiltonian systems whosetrajectories go to infinity in the phase space in a finite time.An extension of the phase space is defined and extensions of thesymplectic form and the Hamiltonian function to the extended spaceare given. The trajectories of the extended Hamiltonian systemdetermine extensions of solutions of the original Hamiltonianequations through the moment when they go to infinity. TheHamiltonian flow in the extended phase space admits a unitaryrepresentation using the Koopman group. Subspaces of strongcontinuity of the unitary representation are found and propertiesof its generator are described.</p>

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Analysis of Exploding Solutions of an Infinite-Dimensional Linear Hamiltonian System in Phase Space Extensions

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • V. A. Glazatov,
  • V. Zh. Sakbaev

摘要

Abstract

We study Hamiltonian flows in a real separable Hilbert spaceendowed with a symplectic structure. Measures on the Hilbert spacethat are invariant with respect to the flows of completelyintegrable Hamiltonian systems are investigated. Invariantmeasures are used to study linear Hamiltonian systems whosetrajectories go to infinity in the phase space in a finite time.An extension of the phase space is defined and extensions of thesymplectic form and the Hamiltonian function to the extended spaceare given. The trajectories of the extended Hamiltonian systemdetermine extensions of solutions of the original Hamiltonianequations through the moment when they go to infinity. TheHamiltonian flow in the extended phase space admits a unitaryrepresentation using the Koopman group. Subspaces of strongcontinuity of the unitary representation are found and propertiesof its generator are described.