  We consider a perturbed integrable system with one frequency, and the
approximate dynamics for the actions given by averaging over the angle. The
classical theory grants that, for a perturbation of order epsilon, the error of
this approximation is O(epsilon) on a time scale O(1/epsilon), for epsilon ->
0. We replace this qualitative statement with a fully quantitative estimate; in
certain cases, our approach also gives a reliable error estimate on time scales
larger than 1/epsilon. 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
epsilon 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 rapidly found also in cases when the
time scale makes impossible (within reasonable CPU times) or unreliable the
direct solution of the perturbed system.
