= Linear response in action-angle variables
{title2=$\delta f_{\mathbf k}=i\mathbf k\cdot\partial_{\mathbf J}f_0\int_0^t\Phi_{\mathbf k}(t')e^{-i\mathbf k\cdot\boldsymbol\Omega(t-t')}\,dt'$}
Linearizing the <Collisionless Boltzmann equation> around $f_0(\mathbf J)$ in an integrable background <Hamiltonian> $H_0(\mathbf J)$ gives $(\partial_t+i\mathbf k\cdot\boldsymbol\Omega)\delta f_{\mathbf k}=i\mathbf k\cdot\partial_{\mathbf J}f_0\Phi_{\mathbf k}$, with $\boldsymbol\Omega=\partial_{\mathbf J}H_0$. An <integrating factor> gives the displayed zero-initial-perturbation solution. A nonzero initial perturbation adds the freely evolving term $e^{-i\mathbf k\cdot\boldsymbol\Omega t}\delta f_{\mathbf k}(0)$.
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