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This paper consists in a unified exposition of methods and techniques of therenormalization group approach to quantum field theory applied to classicalmechanics, and in a review of results: (1) a proof of the KAM theorem, bystuding the perturbative expansion (Lindstedt series) for the formal solutionof the equations of motion; (2) a proof of a conjecture by Gallavotti about therenormalizability of isochronous hamiltonians, i.e. the possibility to add aterm depending only on the actions in a hamiltonian function not verifying theanisochrony condition so that the resulting hamiltonian is integrable. Suchresults were obtained first by Eliasson; however the difficulties arising inthe study of the perturbative series are very similar to the problems which onehas to deal with in quantum field theory, so that the use the methods whichhave been envisaged and developed in the last twenty years exactly in order tosolve them allows us to obtain unified proofs, both conceptually andtechnically. In the final part of the review, the original work of Eliasson isanalyzed and exposed in detail; its connection with other proofs of the KAMtheorem based on his method is elucidated.Comment: 58, compile with dvips to get the figure

METHODS FOR THE ANALYSIS OF THE LINDSTEDT SERIES FOR KAM TORI AND RENORMALIZABILITY IN CLASSICAL MECHANICS: A review with Some Applications