On Dissipative Phenomena of the Interaction of Hamiltonian Systems

The dynamics is under study of a composite Hamiltonian system that is the union of a finite-dimensional nonlinear system and an infinite-dimensional linear system with quadratic interaction Hamiltonian. The dynamics of the finite-dimensional subsystem is determined by a nonlinear integro-differential equation with a relaxation kernel. We prove existence and uniqueness theorems and find a priori estimates for a solution. Under some assumptions on the form of interaction, the solution to the finite-dimensional subsystem converges to one of the critical points of the effective Hamiltonian.