"Math World" has a brief and interesting discussion of the KAM
theorem and its history. Kolmogorov's original article proposing the KAM idea is hopefully
quite readable: Kolmogorov, A.N. "Preservation of conditionally
periodic movements with small change in the Hamiltonian function".
Dokl. Akad. Nauk. SSSR 98, 527 (1954). Reprinted in MacKay and Meiss
1987: "Hamiltonian dynamical systems: a reprint selection" (Adam
Hilger, Bristol, 1987); you can find it here
A completely different approach is given, for example, in
A lecture by Poschel with the complete and detailed proof of the KAM
theorem, which you can find here.
http://arxiv.org/abs/chao-dyn/9807029
From: Antti Kupiainen (ajkupiai@cc.helsinki.fi)
Date: Tue, 21 Jul 1998 21:31:58 GMT (33kb)
KAM Theorem and Quantum Field Theory
Authors: J.Bricmont, K.Gawedzki, A.Kupiainen
Comments: 32 pages
Subj-class: Chaotic Dynamics; Mathematical Physics
Journal-ref: Commun.Math.Phys. 201 (1999) 699-727
We give a new proof of the KAM theorem for analytic Hamiltonians. The
proof is inspired by a quantum field theory formulation of the problem
and is based on a renormalization group argument treating the small
denominators inductively scale by scale. The crucial cancellations of
resonances are shown to follow from the Ward identities expressing the
translation invariance of the corresponding field theory.