A classical self--contained proof of Kolmogorov's theorem on invariant tori

by Antonio Giorgilli and Ugo Locatelli

**Abstract:**
The celebrated theorem of Kolmogorov on persistence of
invariant tori of a nearly integrable Hamiltonian system is revisited
in the light of classical perturbation algorithm. It is shown that the
original Kolmogorov's algorithm can be given the form of a
constructive scheme based on expansion in a parameter. A careful
analysis of the accumulation of the small divisors shows
that it can be controlled geometrically. As a consequence, the proof
of convergence is based essentially on Cauchy's majorant's method,
with no use of the so called quadratic method. A short comparison with
Lindstedt's series is included.

*Hamiltonian systems with three or more degrees of freedom*,
Carles Simó ed., pp 72-89, NATO ASI series C, Vol. 533, Kluwer Academic Publishers, Dordrecht--Boston--London (1999)

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