= Solution
For a regular <integrable Hamiltonian system>, <action-angle variables> offer two related simplifications.
* Unperturbed <Hamilton's equations> are $\dot{\mathbf J}=0$ and $\dot{\boldsymbol\theta}=\boldsymbol\Omega(\mathbf J)$. Complicated <stellar orbits> become uniform motion on an <invariant torus>, and the unperturbed transport operator becomes $\boldsymbol\Omega\cdot\partial_{\boldsymbol\theta}$. A <Fourier series> in the angles then makes <orbital resonances> explicit through $\mathbf k\cdot\boldsymbol\Omega$.
* 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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