= Solution
For an integrable or approximately integrable background <Hamiltonian>, <action-angle variables> turn an otherwise complicated <stellar orbit> into $\dot{\mathbf J}=0$ and $\dot{\boldsymbol\theta}=\boldsymbol\Omega(\mathbf J)$. This has two useful consequences. First, a phase-mixed equilibrium can be represented directly as $f_0(\mathbf J)$: the <action variables> label the unperturbed <orbits>, while <angle variables> describe position along them. Second, the angles are periodic, so <Fourier series> reduce unperturbed transport to multiplication by $i\mathbf n\cdot\boldsymbol\Omega$ and make <orbital resonances> explicit. The <action variables> also separate rapid orbital motion from slow secular changes. These advantages apply on regular orbital regions; a globally chaotic background need not admit global <action-angle variables>.
The transformation is canonical, preserving the <phase space> measure and the stated total <mass> $M=\int f\,d^3\theta\,d^3J$. The collisionless <Vlasov equation> is
$$
\boxed{\partial_tf+\{f,H\}_{\theta,J}=\partial_tf+\frac{\partial H}{\partial\mathbf J}\cdot\frac{\partial f}{\partial\boldsymbol\theta}-\frac{\partial H}{\partial\boldsymbol\theta}\cdot\frac{\partial f}{\partial\mathbf J}=0.}
$$
Here $H$ is the single-star specific <Hamiltonian>; the normalization of $f$ by <mass> changes neither the <Poisson bracket> nor the transport equation.
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