  In systems where one coordinate undergoes periodic oscillation, the net
displacement in any other coordinate over a single period is shown to be given
by differentiation of the action integral associated with the oscillating
coordinate. This result is then used to demonstrate that the action integral
acts as a Hamiltonian for slow coordinates providing time is scaled to the
``tick-time'' of the oscillating coordinate. Numerous examples, including
charged particle drifts and relativistic motion, are supplied to illustrate the
varied application of these results.
