Theorems and problems in fully coupled averaging
Abstract. Averaging set up arises in the study of systems obtained as a perturbation of systems having integrals of motion. This leads to a combination of a slow and a fast motion and the averaging prescription suggests to approximate the slow motion by the averaged one. When the fast motion does not depend on the slow one this prescription works, essentially, always and a lot is known about the averaging approximation and its error. In the general (fully coupled) case the situation is known to be more complex in view of resonances and the need to consider slowly changing dynamical systems. Some old and new results, as well, as remaining problems concerning this topic will be discussed.