Linear response in action-angle variables ($\delta f_{\mathbf{k}}=i\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0{\int }_0^t{\Phi }_{\mathbf{k}}(t^{\prime })e^{-i\mathbf{k}\cdot \mathbf{\Omega }(t-t^{\prime })}d{t}^{\prime }$)

Linearizing the Collisionless Boltzmann equation around $f_0(\mathbf{J})$ in an integrable background Hamiltonian $H_0(\mathbf{J})$ gives $({\partial }_t+i\mathbf{k}\cdot \mathbf{\Omega })\delta f_{\mathbf{k}}=i\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0{\Phi }_{\mathbf{k}}$, with $\mathbf{\Omega }={\partial }_{\mathbf{J}}{H}_0$. An integrating factor gives the displayed zero-initial-perturbation solution. A nonzero initial perturbation adds the freely evolving term $e^{-i\mathbf{k}\cdot \mathbf{\Omega }t}\delta f_{\mathbf{k}}(0)$.