Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-04T10:21:52.712Z Has data issue: false hasContentIssue false

Some remarks on the classical KAM theorem, following Pöschel

Published online by Cambridge University Press:  10 May 2024

Albert Fathi
Affiliation:
Georgia Institute of Technology
Philip J. Morrison
Affiliation:
University of Texas, Austin
Tere M-Seara
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Sergei Tabachnikov
Affiliation:
Pennsylvania State University
Get access

Summary

We propose a slight correction and a slight improvement on the main result contained in A Lecture on Classical KAM Theorem by J. Pöschel.

Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
, pp. 67 - 74
Publisher: Cambridge University Press
Print publication year: 2024

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

Bounemoura, A., “Positive measure of KAM tori for finitely differentiable Hamiltonians”, J. Éc. polytech. Math. 7 (2020), 11131132.CrossRefGoogle Scholar
Bounemoura, A. and Féjoz, J., “KAM, α-Gevrey regularity and the α-Bruno–Rüssmann condition”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19:4 (2019), 12251279.Google Scholar
Koudjinan, C. E., “Quantitative KAM normal forms and sharp measure estimates”, preprint, 2019. arXiv 1904.13062Google Scholar
Popov, G., “KAM theorem for Gevrey Hamiltonians”, Ergodic Theory Dynam. Systems 24:5 (2004), 17531786.CrossRefGoogle Scholar
Pöschel, J., “A lecture on the classical KAM theorem”, pp. 707732 in Smooth ergodic theory and its applications (Seattle, WA, 1999), edited by Katok, A. et al., Proc. Sympos. Pure Math. 69, Amer. Math. Soc., Providence, RI, 2001.Google Scholar
Rüssmann, H., “Nondegeneracy in the perturbation theory of integrable dynamical systems”, pp. 518 in Number theory and dynamical systems (York, 1987), edited by Dodson, M. M. and Vickers, J. A. G., Math, London. Soc. Lecture Note Ser. 134, Cambridge Univ. Press, 1989.Google Scholar
Rüssmann, H., “Invariant tori in non-degenerate nearly integrable Hamiltonian systems”, Regul. Chaotic Dyn. 6:2 (2001), 119204.CrossRefGoogle Scholar
Rüssmann, H., “Invariant tori in non-degenerate nearly integrable Hamiltonian systems”, Regul. Chaotic Dyn. 6:2 (2001), 119204.CrossRefGoogle Scholar
Villanueva, J., “Kolmogorov theorem revisited”, J. Differential Equations 244:9 (2008), 22512276.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×