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Current theories of dynamical friction on galactic bars are based either onlinear perturbation theory, which is valid only in the fast limit where the barchanges its pattern speed rapidly, or on adiabatic theory, which is applicableonly in the slow limit where the bar's pattern speed is near-constant. In thispaper, we study dynamical friction on galactic bars spinning down at anarbitrary speed, seamlessly connecting the fast and slow limits. We treat thebar-halo interaction as a restricted $N$-body problem and solve thecollisionless Boltzmann equation using the fast-angle-averaged Hamiltonian. Thephase-space distribution and density wakes predicted by our averaged model arein excellent agreement with full 3D simulations. In the slow regime whereresonant trapping occurs, we show that, in addition to the frictional torque,angular momentum is transferred directly due to the migration of the trappedphase-space: trapped orbits comoving with the resonance typically gain angularmomentum, while untrapped orbits leaping over the trapped island lose angularmomentum. Due to the negative gradient in the distribution function, gainerstypically outnumber the losers, resulting in a net negative torque on theperturber. Part of this torque due to the untrapped orbits was alreadyidentified by Tremaine & Weinberg who named the phenomenon dynamical feedback.Here, we derive the complete formula for dynamical feedback, accounting forboth trapped and untrapped orbits. Using our revised formula, we show thatdynamical feedback can account for up to $30\%$ of the total torque on theMilky Way's bar.

Dynamical friction and feedback on galactic bars in the general fast-slow regime