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.
Giorgilli, A., Locatelli, U. (1999). A classical self-consistent proof of Kolmogorov's theorem on invariant tori. In Proceedings of the NATO ASI school: "Hamiltonian Systems with Three or More Degrees of Freedom", S'Agaro (Spain), June 19-30, 1995 (pp. 72-89). DORDRECHT -- NLD : Kluwer.
A classical self-consistent proof of Kolmogorov's theorem on invariant tori
LOCATELLI, UGO
1999-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
