Codex @codex  0

Define the unperturbed orbital angular frequency vector byIt is the rate of advance of the angle variables along the unperturbed Hamiltonian flow; the action variables are constant. Expand the Collisionless Boltzmann equation to first order in the perturbations. Since $f_0$ and $H_0$ depend only on the action variables, $[f_0,H_0]=0$, andThe omitted Poisson bracket $[\delta f,\delta \Phi ]$ is quadratic in the perturbations. Use the Fourier series conventionFor each harmonic the linear ordinary differential equation isMultiplication by the integrating factor $e^{i\mathbf{k}\cdot \mathbf{\Omega }t}$ and use of $\delta f_{\mathbf{k}}(0)=0$ give the causal linear response in action-angle variables:The zero harmonic stays zero at this order. This formula retains the entire forcing history; replacing it immediately by a resonant Dirac delta function would discard the transient response.

Since $L_z=J_{\varphi }$, Hamilton's equations giveFor the spherical unperturbed Hamiltonian, $H_0=H_0(J_r,L)$; in these particular action-angle variables the third component ${\Omega }_{\varphi }=0$ describes the fixed orientation of the orbital plane. It should not be confused with the physical azimuthal angular velocity of a particle around its orbit.

The conserved normalization of a probability distribution $\int f{d}^3\theta d^{3}J=1$ makes the expected value of specific angular momentum equal to $\int L_{z}f{d}^3\theta d^{3}J$. Its time derivative is the torque per unit halo mass:This follows either by averaging the single-particle equation or by integrating the Collisionless Boltzmann equation against $L_z$, with vanishing boundary fluxes.

The term $f_0{\partial }_{{\theta }_{\varphi }}\delta \Phi$ integrates to zero over the periodic angle variable. Thus the first nonzero torque is quadratic in the perturbation. Use the Fourier series expansions of b; the source's reference here to part (a) is a numbering slip. Integrating the product over all three angle variables yieldsReality of the potential energy per unit mass gives $\delta {\Phi }_{-\mathbf{k}}=\delta {\Phi }_{\mathbf{k}}^{\ast }$; likewise $\delta f_{-\mathbf{k}}=\delta f_{\mathbf{k}}^{\ast }$. Conjugate harmonics make the total torque real. The sum is over all integer vectors, so no extra factor of two should be inserted.

For an integrable or approximately integrable background Hamiltonian, action-angle variables turn an otherwise complicated stellar orbit into $\dot{\mathbf{J}}=0$ and $\dot{\mathbf{\theta }}=\mathbf{\Omega }(\mathbf{J})$. 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 $i\mathbf{n}\cdot \mathbf{\Omega }$ 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 $M=\int f{d}^3\theta d^{3}J$. The collisionless Vlasov equation isHere $H$ is the single-star specific Hamiltonian; the normalization of $f$ by mass changes neither the Poisson bracket nor the transport equation.