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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