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 majorants method, with no use of the so called quadratic method. A short comparison with Lindstedt’s series is included.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kolmogorov, A.N.: Preservation of conditionally periodic movements with small change in the Hamilton function, Dokl. A.ad. Nauk SSSR, 98, 527 (1954).
Moser, J.: Convergent series expansions for quasi-periodic motions, Math. Ann. 169, 136–176 (1967).
Eliasson, L. H.: Absolutely convergent series expansion for quasi-periodic motions, report 2 88, Dept. of Math., Univ. of Stockolm (1988).
Eliasson, L. H.: Hamiltonian system with normal form near an invariant torus, in Nonlinear dynamics, G. Turchetti ed., World scientific (1989).
Eliasson, L. H.: Generalization of an estimate of small divisors by Siegel, in P. Rabinowitz and E. Zehnder eds., book in honor of J. Moser, Academic Press (1990).
Gallavotti, G.: Twistless KAM tori, quasi fiat homoclinic intersections, and other cancellations in the perturbation series of certain completely mtegrable Hamiltonian systems. A review, Reviews in Math. Phys. 6, 343–411 (1994).
Gallavotti, G.: Twistless KAM ton, Comm. Math. Phys. 164, 145–156 (1994).
Chierchia, L. and Falcolini, C.: A direct proof of a theorem by Kolmogorov in Hamiltonian systems, preprint (1993).
Gallavotti, G. and Gentile, G.: Nonrecursive proof of the KAM theorem, preprint (1994).
Gentile, G. and Mastropietro, V.: Tree expansion and multiscale analysis for KAM tori, preprint (1994).
Giorgilli, A. and Locatelli, U.: Kolmogorov theorem and classical perturbation theory, preprint (1995).
Gröbner, W.: Die Lie-Reihen und Ihre Anwendungen, Springer Verlag, Berlin (1960); it. transi.: Le serie di Lie e le loro applicazioni, Cremonese, Roma (1973).
Rüssmann, H.: Über die Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Annalen, 169, 55–72 (1967).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Giorgilli, A., Locatelli, U. (1999). A Classical Self-Contained Proof of Kolmogorov’s Theorem on Invariant Tori. In: Simó, C. (eds) Hamiltonian Systems with Three or More Degrees of Freedom. NATO ASI Series, vol 533. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4673-9_8
Download citation
DOI: https://doi.org/10.1007/978-94-011-4673-9_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5968-8
Online ISBN: 978-94-011-4673-9
eBook Packages: Springer Book Archive