Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics

This paper consists in a unified exposition of methods and techniques of the renormalization group approach to quantum field theory applied to classical mechanics, and in a review of results: (1) a proof of the KAM theorem, by studying the perturbative expansion (Lindstedt series) for the formal solution of the equations of motion; (2) a proof of a conjecture by Gallavotti about the renormalizability of isochronous hamiltonians, i.e. the possibility to add a term depending only on the actions in a hamiltonian function not verifying the anisochrony condition so that the resulting hamiltonian is integrable. Such results were obtained first by Eliasson; however the difficulties arising in the study of the perturbative series are very similar to the problems which one has to deal with in quantum field theory, so that the use of the methods which have been envisaged and developed in the last twenty years precisely in order to solve them allows us to obtain unified proofs, both conceptually and technically. In the final part of the review, the original work of Eliasson is analyzed and exposed in detail; its connection with other proofs of the KAM theorem based on his method is elucidated.