Many problems in dynamics are described by well ordered motions subjet to
periodic (or quasi-periodic perturbations). An important example will be
the motion of a planet around the sun (according to Kepler laws) subject to
the perturbations of the other planets.
One problem of fundamental practical importance is whether the small
perturbations will accumulate over time or just average out.
The answer turns out to be rather complicated and depends on delicate
number theoretic properties and regularity issues. Nevertheless, it is
possible to get a geometric understanding of where the instability occurs.
In some cases, one can take advantage of this accumulation of perturbations
to make complicated maneouvers of spacecraft using the tides of the
planets, which are free instead of expensive fuel.
