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On dissipative phenomena of the interaction of Hamiltonian systems. (Russian, English) Zbl 1031.37045

Sib. Mat. Zh. 44, No. 1, 73-86 (2003); translation in Sib. Math. J. 44, No. 1, 61-72 (2003).
The dynamics of a Hamiltonian system of composite type, which is the union of a finite-dimensional nonlinear system and an infinite-dimensional linear system with quadratic Hamiltonian of interaction are studied. The author shows that the dynamics of the finite-dimensional subsystem is determined by a nonlinear integro-differential equation with the so-called relaxation kernel. Existence and uniqueness theorems for a solution are presented. The technique of a priori estimates is used for proving the existence of a global solution. The author shows that, under some conditions on the form of the interaction, the solution obtained for the finite-dimensional system converges to a critical point of an effective Hamiltonian.

MSC:

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
45K05 Integro-partial differential equations
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