Normal stability of slow manifolds in nearly-periodic Hamiltonian systems

TL;DR

This paper proves the existence and stability of slow manifolds in nearly-periodic Hamiltonian systems, providing conditions for long-term stability and applications to guiding center dynamics in plasma physics.

Contribution

It introduces nearly-invariant slow manifolds in nearly-periodic Hamiltonian systems and establishes their long-term normal stability, extending the understanding of these systems.

Findings

Existence of nearly-invariant slow manifolds in such systems

A sufficient condition for long-term stability based on adiabatic invariants

Validation of stable embeddings for guiding center dynamics

Abstract

M. Kruskal 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.