= Resonant torque from a rotating perturbation
{title2=$\mathcal T=-\pi(2\pi)^3\sum_{\mathbf k}m\int|\Psi_{\mathbf k}|^2\mathbf k\cdot\partial_{\mathbf J}f_0\,\delta(\mathbf k\cdot\boldsymbol\Omega-m\Omega_p)\,d^3J$}
The leading second-order specific <torque> on a collisionless distribution from a weak steadily rotating perturbation with $\Phi_{\mathbf k}=\Psi_{\mathbf k}e^{-im\Omega_pt}$ and $m=k_\phi$. The sum includes conjugate harmonics. <Linear response in action-angle variables> gives the finite-time kernel $\sin(\Delta t)/\Delta$; its integrated <long-time resonance kernel> selects orbital <resonances>. This limit assumes a smooth continuum, regular resonant surfaces and negligible secular redistribution during the response calculation.
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