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For a regular integrable Hamiltonian system, action-angle variables offer two related simplifications.

These advantages assume regular or approximately integrable mean motion; globally defined action-angle variables need not exist in chaotic regions.

The required time separation is $\tau \ll t_{\sec }$, where $t_{\sec }$ is the evolution timescale of $f_0$, together with $t_{\mathrm{peak}}\gg \tau$ and an observation time well after the pulse. The first condition permits freezing $f_0(\mathbf{J},t^{\prime })$ at its pre-pulse value; its neglected change would enter beyond the retained second order. The second makes the original initial time lie in a negligible tail of the Gaussian function, permitting replacement of the lower time limits by $-\infty$. Short duration alone would not justify that replacement for a pulse centered at the initial time.

Integrate the continuity equation in c over the whole pulse. For fixed $\mathbf{J}$, the time factor is the ordered Gaussian response integralA common translation by $t_{\mathrm{peak}}$ removes the peak time. Pair the positive and negative azimuthal harmonics: $(-m,-k_R)$ is paired with $(m,k_R)$. Both $k_{\phi }$ and $\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0$ change sign, so their product does not; ${\omega }_{-\mathbf{k}}=-{\omega }_{\mathbf{k}}$ makes the two time factors complex conjugates. Hence the pair contributes $2\mathrm{Re}\mathcal{I}$.

For completeness, the real part of the integrand is symmetric under $t\leftrightarrow t^{\prime }$. The triangular time domain therefore gives half of the full plane, and the Fourier transform of a Gaussian function givesThis derives the supplied identity rather than requiring a separate time-ordering calculation.

DefineUsing the paired harmonics and $\Delta F_0=-{\partial }_{J_{\phi }}\int Q_{0}dt$ now yieldsBy the stated assumption, ${\omega }_{m{k}_R}$ is independent of $J_R$, so its exponential can be taken outside the radial integral. Put $\Gamma =(\sqrt{2}\tau {)}^{-1}$ and useThereforeThe outer derivative acts on the Gaussian resonance broadening factor as well as on the radial integral. The result describes quasilinear diffusion in action space weighted by the squared pulse amplitude, with a detuning width $\Gamma$. A shorter pulse has broader frequency selectivity, while a longer pulse selects narrower orbital resonances. If the action-space boundary flux vanishes, integrating $\Delta F_0$ over $J_{\phi }$ gives zero, as required by conservation of mass.

The diffusive evolution of an averaged phase-space distribution function caused by weak fluctuating fields in the quasilinear approximation. In a stationary homogeneous one-dimensional electrostatic problem, if the potential's temporal power spectral density is $S_{\phi ,k}(\omega )$ with covariance $\int S_{\phi ,k}(\omega )e^{-i\omega \tau }d\omega /(2\pi )$, thenThis follows from the real part of the one-sided time correlation along an unperturbed particle trajectory. Both signs of the spatial wavenumber are included. The action-space counterpart for regular stellar orbits is quasilinear diffusion in action space.