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On phase at a resonance in slow-fast Hamiltonian systems

journal contribution
posted on 2023-09-18, 09:03 authored by Yuyang GaoYuyang Gao, Anatoly NeishtadtAnatoly Neishtadt, Alexey Okunev

We consider a slow-fast Hamiltonian system with one fast angular variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles in inhomogeneous magnetic field under influence of a high-frequency electrostatic waves. Trajectories of the averaged over the fast phase system cross the resonant surface. The fast phase makes ∼ 1/ε turns before arrival to the resonant surface (ε is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival to the resonance was derived earlier in the context of study of charged particle dynamics on the basis of heuristic considerations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove that its accuracy is O(√ε) (up to a logarithmic correction). This estimate for the accuracy is optimal.

Funding

Leverhulme Trust (Grant No. RPG-2018-143)

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Regular and Chaotic Dynamics

Volume

28

Issue

4-5

Pages

581 - 608

Publisher

Springer

Version

  • AM (Accepted Manuscript)

Publisher statement

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/[insert DOI]

Acceptance date

2023-06-19

ISSN

1560-3547

eISSN

1468-4845

Language

  • en

Depositor

Prof Anatoly Neishtadt. Deposit date: 14 September 2023

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