  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.
