Guido Gentile, Vieri Mastropietro
Methods for the analysis of the Lindstedt series for KAM tori
and renormalizability in classical mechanics.
A review with some applications.
(181K, Plain Tex)
ABSTRACT. 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,
\ie 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.