American Institute of Mathematical Sciences

Advanced
February  2004, 10(1&2): 367-385. doi: 10.3934/dcds.2004.10.367

A proof of Kolmogorov's theorem

John Hubbard 1, and  Yulij Ilyashenko 1,

1. 

Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853, United States, United States

Received  May 2002 Revised  April 2003 Published  October 2003

In this paper we will give a proof of Kolmogorov's theorem on the conservation of invariant tori. This proof is close to the one given by Bennettin, Galgani, Giorgilli and Strelcyn in [2]; we follow the outline of their proof, but carry out the steps somewhat differently in several places. In particular, the use of balls rather than polydiscs simplifies several arguments and improves the estimates.
Keywords: Hamiltonian mechanics, Invariant tori, KAM theory.
Mathematics Subject Classification: Primary: 70H08, 70H07, 70H0.
Citation: John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367
[1]

Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371
[2]

Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633
[3]

C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457
[4]

Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63
[5]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
[6]

Luca Biasco, Luigi Chierchia. On the measure of KAM tori in two degrees of freedom. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6635-6648. doi: 10.3934/dcds.2020134
[7]

Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941
[8]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[9]

Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120
[10]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[11]

Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162
[12]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[13]

Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377
[14]

Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147
[15]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[16]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[17]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[18]

Eduard Feireisl, Šárka Nečasová, Reimund Rautmann, Werner Varnhorn. New developments in mathematical theory of fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : i-ii. doi: 10.3934/dcdss.2014.7.5i
[19]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092
[20]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (115)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy