  In the vinicity of phase-space resonances for a given species of plasma, the
particle distribution function is flattened as free energy is exchanged between
the plasma and resonant electromagnetic waves. Here, we present action-angle
variables which explicitly separate the adiabatic invariant from the
contribution that arises from perturbation of the Hamiltonian over the course
of a single orbit. Then, we perform similar analysis for the orbit period,
allowing one to identify a generating function $\psi$ which modifies the zeroth
order contribution to the orbit period (unperturbed orbit) in the case where
the particle energy is not time-invariant. We posit that a population of
`quasi-trapped' particles cross the separatrix, directly allowing for island
growth and decay. Then, we employ $\psi$ in the ensemble case by considering
how this generating function allows for solutions of the 6+1D electrostatic
Vlasov equation where finite growth and frequency sweeping of modes occur.
Finally, we derive an approximate form for $\psi$ outside of the separatrix,
allowing for qualitative observation of phase-space shear and island formation.
