On the average principle for one-frequency systems

We consider a perturbed integrable system with one frequency, and the approximate dynamics for the actions given by averaging over the angle. A classical qualitative result states that, for a perturbation of order ε, the error of this approximation is O(ε) on a time scale O(1/ε), for ε → 0. We replace this with a fully quantitative estimate; in certain cases, our approach also gives a reliable error estimate on time scales larger than 1/ε. A number of examples are presented; in many cases, our estimator practically coincides with the envelope of the rapidly oscillating distance between the actions of the perturbed and of the averaged systems. Fairly good results are also obtained in some 'resonant' cases, where the angular frequency is small along the trajectory of the system. Even though our estimates are proved theoretically, their computation in specific applications typically requires the numerical solution of a system of differential equations. However, the time scale for this system is smaller by a factor ε than the time scale for the perturbed system. For this reason, computation of our estimator is faster than the direct numerical solution of the perturbed system; the estimator is also rapidly found in the cases when the time scale makes impossible (within reasonable CPU times) or unreliable the direct solution of the perturbed system.