  The shape of galaxies depends on their orbital populations. These populations
change through capture into and escape from resonance. Capture problems fall
into distinct cases depending upon the shape of the potential well. To
visualise the effective potential well for orbital capture, a diagrammatic
approach to the resonant perturbation theory of Born is presented. These
diagrams we call equiaction sections. To illustrate their use, we present
examples drawn from both galactic and Solar System dynamics. The probability of
capture for generic shapes of the potential well is calculated. A number of
predictions are made. First, there are barred galaxies that possess two outer
rings of gas and stars (type R1R2). We show how to relate changes in the
pattern speed and amplitude of the bar to the strength of the two rings.
Secondly, under certain conditions, small disturbances can lead to dramatic
changes in orbital shape. This can be exploited as a mechanism to pump
counter-rotating stars and gas into the nuclei of disk galaxies. Tidal resonant
forcing of highly inclined orbits around a central mass causes a substantial
increase in the likelihood of collision. Thirdly, the angular momentum of a
potential well is changed by the passage of stars across or capture into the
well. This can lead to the creation of holes, notches and high velocity tails
in the stellar distribution function, whose form we explicitly calculate.
