Smoothed Langevin dynamics of highly oscillatory systems

We consider Hamiltonian systems containing a strong constraining potential. Such systems were first investigated by Rubin and Ungar who derived the corresponding reduced equations of motion for the slow degree of freedom. The purpose of this paper is two-fold. (i) We transform the highly oscillatory degree of freedom to action–angle variables and use recent results on the truncation error of the corresponding normal form transformations to derive an estimate on the lifespan of the underlying adiabatic invariant. (ii) We investigate thermally embedded Hamiltonian systems, i.e. Langevin dynamics, containing a strong constraining potential. Using again a transformation to action–angle variables and applying the principle of averaging, we derive reduced equations of motion for the slowly varying degrees of freedom. This includes in particular the derivation of a correcting force-term that stands for the coupling of the slow and fast degrees of motion. In case of strong thermal coupling, our reduced equations of motion suggest that the correcting force term can be approximated by the gradient of the Fixman potential as used in statistical mechanics. We also discuss smoothing in the context of constant temperature molecular dynamics and provide a numerical example.