Introduction to Perturbative Methods

In this final Chapter we outline the basic facts and techniques of perturbation theory, mostly without proofs. After a general introduction to the problem, we briefly discuss time-dependent perturbative methods. Then we focus on time-independent perturbation theory. We discuss the fundamental equation of perturbation theory and the problem of “small denominators”. To solve this problem we introduce the Diophantine conditions on the frequencies. We discuss the relation of convergence of the perturbative series with the existence of enough conserved quantities and give some negative results. The last three sections are dedicated, respectively, to the KAM theory, to the Nekhoroshev theorem, and to the Birkhoff normal forms around a point of absolute equilibrium. In the appendix we discuss the Diophantine condition and Liouville numbers.