  Lynden-Bell & Kalnajs (1972) presented a useful formula for computing the
long-range torque between spiral arms and the disk at large. The derivation
uses second-order perturbation theory and assumes that the perturbation slowly
grows over a very long time: the time-asymptotic limit. This formula has been
widely used to predict the angular momentum transport between spiral arms and
stellar bars between disks and dark-matter halos. However, this paper shows
that the LBK time-asymptotic limit is not appropriate because the
characteristic evolution time for galaxies is too close to the relevant
dynamical times. We demonstrate that transients, not present in the
time-asymptotic formula, can play a major role in the evolution for realistic
astronomical time scales. A generalisation for arbitrary time dependence is
presented and illustrated by the bar--halo and satellite--halo interaction. The
natural time dependence in bar-driven halo evolution causes quantitative
differences in the overall torque and qualitative differences in the physical-
and phase-space location of angular momentum transfer. The time-dependent
theory predicts that four principal resonances dominate the torque at different
times and accurately predicts the results of an N-body simulation. In addition,
we show that the Inner Lindblad Resonance (ILR) is responsible for the peak
angular momentum exchange but, due to the time dependence, the changes occur
over a broad range of energies, radii and frequencies. We describe the
implication of these findings for the satellite--halo interaction using a
simple model and end with a discussion of possible impact on other aspects
secular galaxy evolution.
