Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-12T19:57:04.668Z Has data issue: false hasContentIssue false
  • English
  • Français

The KAM theorem and renormalization group

Published online by Cambridge University Press:  01 April 2009

E. DE SIMONE  and
A. KUPIAINEN
E. DE SIMONE
Affiliation:
Department of Mathematics and Statistics, PO Box 68 (Gustaf Hällströmin katu 2b) Helsinki, 00014, Finland (email: emiliano.desimone@helsinki.fi)
A. KUPIAINEN
Affiliation:
Department of Mathematics and Statistics, PO Box 68 (Gustaf Hällströmin katu 2b) Helsinki, 00014, Finland (email: emiliano.desimone@helsinki.fi)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a certain PDE with quadratic nonlinearity, the so-called Polchinski renormalization group equation studied in quantum field theory.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 419 - 431
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Kolmogorov, A. N.. On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98(4) (1954), 527–530 (in Russian).Google Scholar
[2]Arnold, V. I.. Proof of A. N. Kolmogorov’s theorem on the preservation of quasi-periodic motions under small perturbation of the Hamiltonian. Uspekhi Mat. Nauk 18(5) (1963), 13–40 (in Russian) (Engl. transl. Russian Math. Surveys 18(5) (1963), 9–36).Google Scholar
[3]Moser, J.. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II a, Nr. I (1962), 1–20.Google Scholar
[4]Eliasson, L. H.. Absolutely convergent series expansions for quasi periodic motions. Reports Department of Math., no. 2, 1–31, University of Stockholm, Sweden, 1988. Published in Math. Phys. Electron. J. 2(4) (1996) 1–33 (http://www.ma.utexas.edu/mpej/).Google Scholar
[5]Gallavotti, G.. Twistless KAM tori. Comm. Math. Phys. 164 (1994), 145–156.Google Scholar
[6]Gallavotti, G.. Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review. Rev. Math. Phys. 6 (1994), 343–411.CrossRefGoogle Scholar
[7]Gallavotti, G.. Invariant tori: a field theoretic point of view on Eliasson’s work. Advances in Dynamical Systems and Quantum Physics. Ed. R. Figari. World Scientific, Singapore, 1995, pp. 117–132.Google Scholar
[8]Chierchia, L. and Falcolini, C.. A direct proof of a theorem by Kolmogorov in Hamiltonian systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 21(4) (1994), 541–593.Google Scholar
[9]Chierchia, L. and Falcolini, C.. Compensations in small divisor problems. Comm. Math. Phys. 175 (1996), 135–160.Google Scholar
[10]Bonetto, F., Gallavotti, G., Gentile, G. and Mastropietro, V.. Quasi linear flows on tori: regularity of their linearization. Comm. Math. Phys. 192 (1998), 707–736.Google Scholar
[11]Gentile, G. and Mastropietro, V.. KAM theorem revisited. Phys. D 90(3) (1996), 225–234.CrossRefGoogle Scholar
[12]Bricmont, J., Gawȩdzki, C. and Kupiainen, A.. KAM theorem and quantum field theory. Comm. Math. Phys. 201(3) (1999), 699–727.CrossRefGoogle Scholar
[13]Bricmont, J., Kupiainen, A. and Schenkel, A.. Renormalization group and the Melnikov problem for PDE’s. Comm. Math. Phys. 221 (2001), 101–140.CrossRefGoogle Scholar
[14]Polchinski, J.. Renormalization of effective Lagrangians. Nuclear Phys. B 231 (1984), 269–295.CrossRefGoogle Scholar
[15]Kadanoff, L. P.. Scaling for a critical Kolmogorov–Arnold–Moser trajectory. Phys. Rev. Lett. 47 (1981), 1641–1643.CrossRefGoogle Scholar
[16]Shenker, S. J. and Kadanoff, L. P.. Critical Behaviour of a KAM surface. I. Empirical Results. J. Stat. Phys. 27 (1982), 631–656.Google Scholar
[17]Stenlund, M.. Construction of whiskers for the quasiperiodically forced pendulum. Rev. Math. Phys. 19(8) (2007), 823–877.Google Scholar
[18]De Simone, E.. A renormalization proof of the KAM theorem for non-analytic perturbations. Rev. Math. Phys. 19(6) (2007), 639–675.Google Scholar
[19]Koch, H.. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. 11(4) (2004), 881–909.Google Scholar