Normal stability of slow manifolds in nearly periodic Hamiltonian systems

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2021-09-01
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en
Pages
27
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Journal of Mathematical Physics, Volume 62, issue 9
Abstract
Kruskal [J. Math. Phys. 3, 806 (1962)] showed that each nearly periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly periodic Hamiltonian systems on barely symplectic manifolds—manifolds equipped with closed, non-degenerate 2-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly periodic system. We prove that one previous embedding and two new embeddings enjoy long-term normal stability and thereby strengthen the theoretical justification for these models.
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Funding Information: The work of J.W.B. was supported by the Los Alamos National Laboratory LDRD program under Project No. 20180756PRD4. The work of E.H. was supported by the Academy of Finland (Grant No. 315278). Any subjective views or opinions expressed herein do not necessarily represent the views of the Academy of Finland or Aalto University. Publisher Copyright: © 2021 Author(s).
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Burby , J W & Hirvijoki , E 2021 , ' Normal stability of slow manifolds in nearly periodic Hamiltonian systems ' , Journal of Mathematical Physics , vol. 62 , no. 9 , 093506 . https://doi.org/10.1063/5.0054323