Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom

We consider the problem of potential interaction between a finite-dimensional linear Lagrangian system and an infinite-dimensional one (a system of linear oscillators and a thermostat). We study the final dynamics of the system. Under natural assumptions, this dynamics turns out to be very simple and admits an explicit description because the thermostat produces an effective dissipation despite the energy conservation and the Lagrangian nature of the system. We use the methods of [1], where the final dynamics of the finite-dimensional subsystem is studied in the case when it has one degree of freedom and a linear potential or (under additional assumptions) polynomial potential. We consider the case of finite-dimensional subsystems with arbitrarily many degrees of freedom and a linear potential and study the final dynamics of the system of oscillators and the thermostat. The necessary assertions from [1] are given with proofs adapted to the present situation.