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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 studing 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 the methods which have been envisaged and developed in the last twenty years exactly 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.

Abstract. Quasi-periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exists after small perturbations. The parametric equations of the invariant tori can often be computed as formal power series in the perturbation parameter and can be given a meaning via resummations. Here we prove that, for a class of elliptic tori, a resummation algorithm can be devised and proved to be convergent, thus extending to such lower-dimensional invariant tori the methods employed to prove convergence of the Lindstedt series either for the maximal (i.e. KAM) tori or for the hyperbolic lower-dimensional invariant tori.

Abstract. We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistorinductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.