Normal stability of slow manifolds in nearly periodic Hamiltonian systems

Burby, Joshua William; Hirvijoki, Eero

doi:10.1063/5.0054323

Normal stability of slow manifolds in nearly periodic Hamiltonian systems

Journal Article · Mon Sep 13 00:00:00 EDT 2021 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/5.0054323· OSTI ID:1828720

^[1]; Hirvijoki, Eero ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Aalto Univ. (Finland)

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. Here, we prove that one previous embedding and two new embeddings enjoy long-term normal stability and thereby strengthen the theoretical justification for these models.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: LDRD

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1828720

Report Number(s):: LA-UR--21-23132

Journal Information:: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 9 Vol. 62; ISSN 0022-2488

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

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Normal stability of slow manifolds in nearly periodic Hamiltonian systems

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