A direct proof of a theorem by Kolmogorov in hamiltonian systems

L. Chierchia; C. Falcolini

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1994)

  • Volume: 21, Issue: 4, page 541-593
  • ISSN: 0391-173X

How to cite

top

Chierchia, L., and Falcolini, C.. "A direct proof of a theorem by Kolmogorov in hamiltonian systems." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 21.4 (1994): 541-593. <http://eudml.org/doc/84192>.

@article{Chierchia1994,
author = {Chierchia, L., Falcolini, C.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Kolmogorov's theorem; quasi-periodic solutions; nearly integrable; Hamiltonian systems; Siegel's method},
language = {eng},
number = {4},
pages = {541-593},
publisher = {Scuola normale superiore},
title = {A direct proof of a theorem by Kolmogorov in hamiltonian systems},
url = {http://eudml.org/doc/84192},
volume = {21},
year = {1994},
}

TY - JOUR
AU - Chierchia, L.
AU - Falcolini, C.
TI - A direct proof of a theorem by Kolmogorov in hamiltonian systems
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1994
PB - Scuola normale superiore
VL - 21
IS - 4
SP - 541
EP - 593
LA - eng
KW - Kolmogorov's theorem; quasi-periodic solutions; nearly integrable; Hamiltonian systems; Siegel's method
UR - http://eudml.org/doc/84192
ER -

References

top
  1. [1] V.I. Arnold, Proof of A.N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbation of the Hamiltonian. Uspekhi Mat. Nauk18, No. 5 (1963), 13-40 (Russian); English translation: Russian Math. Surveys18, No. 5 (1963), 9-36. Zbl0129.16606MR163025
  2. [2] V.I. Arnold, Small denominators and problems of stability of motions in classical and celestial mechanics. Uspekhi Mat. Nauk18, No. 6 (1963), 91-192 (Russian); English translation: Russian Math. Surveys18, No. 6 (1963), 85-192. Zbl0135.42701MR170705
  3. [3] A. Berretti - L. Chierchia, On the complex analytic structure of the golden-mean invariant curve for the standard map. Nonlinearity3 (1990), 39-44. Zbl0732.58014MR1036436
  4. [4] A. Berretti - A. Celletti - L. Chierchia - C. Falcolini, Natural boundaries for area-preserving twist maps. J. Statist. Phys., 66 (1992), 1613-1630. Zbl0892.58062MR1156418
  5. [5] B. Bollobas, Graph Theory. Springer-Verlag (Graduate text in mathematics, 63), Berlin-Heidelberg-New York, 1979. Zbl0411.05032MR536131
  6. [6] A.D. Brjuno, Convergence of transformations of differential equations to normal form. Dokl. Akad. Nauk SSSR165 (1965), 987-989; Analytic form of differential equations, Trans. Moscow Math. Soc.25 (1971), 131-288 and 26 (1972), 199-239. Zbl0146.11301MR192098
  7. [7] A. Celetti - L. Chierchia, Construction of analytic KAM surfaces and effective stability bounds, Comm. Math. Phys.118 (1988), 119-161. Zbl0657.58032MR954678
  8. [8] L. Chierchia - G. Gallavotti, Drift and Diffusion in phase space, Preprint (1992). To appear in Ann. Inst. H. Poincaré Phys. Théor. Zbl1010.37039MR1259103
  9. [9] L. Chierchia - P. Perfetti, Second order Hamiltonian equations on T∞ and almost-periodic solutions. Preprint (1992). To appear in J. Differential Equations. Zbl0821.34043
  10. [10] L. Chierchia - E. Zehnder, Asymptotic expansions of quasiperiodic solutions. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 16 (1989), 245-258. Zbl0695.34040MR1041897
  11. [11] L.H. Eliasson, Absolutely convergent series expansions for quasi periodic motions, Reports Department of Math., Univ. of Stockholm, Sweden, No. 2 (1988), 1-31. MR1399458
  12. [12] L.H. Eliasson, Hamiltonian systems with linear normal form near an invariant torus. In "Nonlinear Dynamics", G. Turchetti (Ed.) World Scientific, Singapore, 1989. Zbl0744.58019MR1115677
  13. [13] L.H. Eliasson, Generalization of an estimate of small divisors by Siegel. In "Analysis, et cetera", P.H. Rabinowitz and E. Zehnder (Eds.), Academic Press, 1990. Zbl0703.34010MR1039350
  14. [14] C. Falcolini - R. De La Llave, Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors, J. Statist. Phys.67 (1992). Zbl0892.58066MR1171147
  15. [15] G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable hamiltonian systems. A review. Preprint (1993). Zbl0798.58036MR1305589
  16. [16] G. Gallavotti - G. Gentile, Non recursive proof of the KAM theorem. Preprint (1993). 
  17. [17] I.P. Goulden - D.M. Jackson, Combinatorial Enumeration. Wyley Interscience Series in Discrete Math., 1983. Zbl0519.05001MR702512
  18. [18] F. Harary, Graph Theory. Addison-Wesley, 1969. Zbl0182.57702MR256911
  19. [19] A.N. Kolmogorov, On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR98, No. 4 (1954), 527-530 (Russian). Zbl0056.31502MR68687
  20. [20] J. Moser, On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20. Zbl0107.29301MR147741
  21. [21] J. Moser, Convergent series expansions for quasi-periodic motions. Math. Ann.169 (1967), 136-176. Zbl0149.29903MR208078
  22. [22] J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations, I-II. Ann. Scuola Norm. Super. Pisa Cl. Sci. (3) 20 (1966), 265-315 and 499-535. Zbl0144.18202
  23. [23] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Vols. 1-3. Gauthier-Villars, Paris1892/1893/1899. 
  24. [24] H. Rüssmann, Kleine Nenner. I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1970 (1970), 67-105. Zbl0201.11202MR273156
  25. [25] W.M. Schmidt, Diophantine Approximation, Springer-Verlag (Lecture Notes in Math.785), Berlin-Heidelberg -New York, 1980. Zbl0421.10019MR568710
  26. [26] C.L. Siegel, Iterations of analytic functions, Ann. of Math.43 (1942), 607-612. Zbl0061.14904MR7044
  27. [27] M. Vittot, Lindstedt perturbation series in Hamiltonian mechanics: explicit formulation via a multidimensional Burmann-Lagrange formula, Preprint (1991). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.