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  • A symplectic approach to Arnold diffusion problems

  • Edited by Albert Fathi, Georgia Institute of Technology, Philip J. Morrison, University of Texas, Austin, Tere M-Seara, Universitat Politècnica de Catalunya, Barcelona, Sergei Tabachnikov, Pennsylvania State University
  • Book:
    Hamiltonian Systems
    Published online:
    10 May 2024
    Print publication:
    09 May 2024, pp 229-296

  • 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.