Averaging in systems with constant frequencies

Abstract

We consider an ordinary differential equation describing a system where a rapid rotation of angle variables is slightly perturbed by a slow evolution of the other (“action”) variables. We will show that, for the special case of constant frequencie of rotation, it is possible to approximate for a long time the exact evolution of the slow variables by a solution averaging the influence of the angle variables. Our proof will closely follow that given by Arnol'd for monofrequent systems, but in our case the problem of “almost resonant” frequencies requires some extra work. Applications to, for instance, problems of classical mechanics are obvious.

Keywords:

• Differential equations
• perturbation theory
• averaging methods
• classical mechanics