Dissipation effects in infinite-dimensional Hamiltonian systems.

We show that the potential coupling of classical mechanical systems (an oscillator and a heat bath), one of which (the heat bath) is linear and infinite-dimensional, can provoke energy dissipation in a finite-dimensional subsystem (the oscillator). Under natural assumptions, the final dynamics of an oscillator thus reduces to a tendency toward equilibrium. D. V. Treschev previously obtained results concerning the dynamics of an oscillator with one degree of freedom and a quadratic or (under some additional assumptions) polynomial potential. Later, A. V. Dymov considered the case of a linear oscillator with an arbitrary (finite) number of degrees of freedom. We generalize these results to the case of a heat bath (consisting of several components) and a multidimensional oscillator (either linear or nonlinear).