Solution

Unperturbed Hamilton's equations are $\dot{J} = 0$ and $\dot{θ} = Ω (J)$ . Complicated stellar orbits become uniform motion on an invariant torus, and the unperturbed transport operator becomes $Ω \cdot \partial_{θ}$ . A Fourier series in the angles then makes orbital resonances explicit through $k \cdot Ω$ .
A phase-mixed equilibrium galactic distribution function depends on the action variables rather than the fast angles, consistently with Jeans theorem. Its slow evolution, including quasilinear diffusion in action space, can therefore be described by an angle average in two action variables rather than by resolving every stellar orbit in position and velocity.

These advantages assume regular or approximately integrable mean motion; globally defined action-angle variables need not exist in chaotic regions.

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