The elements of Hamiltonian Perturbation theory

The purpose of this chapter is to introduce in the simplest possible way the “elements” – i.e. the basic facts, ideas, techniques and results – of Hamiltonian perturbation theory. The exposition is oriented to exponential estimates and Nekhoroshev theorem, but a few pages are devoted to KAM theory, too. Some attention is required for the geometric aspects of erturbation theory, which in general are less known than the analytic–arithmetic ones. Geometry plays a basic role, in particular, in the extension of perturbation theory to “degenerate systems” (the perturbed Kepler problem; the fast rotating rigid body). We shall not develop a general theory of these systems, whose study is rather recent in spite of their physical significance, but we shall discuss in some detail one of them, namely the perturbed Euler–Poinsot rigid body. We have no space, unfortunately, for the numerous applications of exponential estimates and Nekhoroshev theory to physical systems, which are rather abundant (for a survey of applications, see [11]). The chapter is mainly addressed to non expert readers, and for this reason, as well as to introduce some language and notations, we begin by a preliminary section where we recall some very elementary facts on Hamiltonian systems and canonical transformations.