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A symplectic approach to Arnold diffusion problems

Published online by Cambridge University Press:  10 May 2024

By
Jean-Pierre Marco
Edited by
Albert Fathi ,
Philip J. Morrison ,
Tere M-Seara  and
Sergei Tabachnikov
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
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Summary

The purpose of this text is to present a symplectic approach to Arnold diffusion problems, that is, the existence of orbits of perturbed integrable systems along which the action variables experience a drift whose length is independent of the size of the perturbation. We choose to focus on the construction of orbits drifting along chains of cylinders, taking for granted the existence of the chains. We however give a rather complete description of these chains, together with some elements on their symplectic features and some main ideas to prove their existence. We adopt the setting introduced by John Mather to prove the Arnold conjecture for perturbations of Tonelli Hamiltonians, which we see as the appropriate one to set out the various (and numerous) problems of the construction, and give some ideas to show how the symplectic approach may enable one to enlarge its scope.

Keywords

Arnold diffusion problems
Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
 , pp. 229 - 296
Publisher: Cambridge University Press
Print publication year: 2024

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