Charged-particle motion in electromagnetic fields having at least one ignorable spatial coordinate.

We give a rigorous derivation of a theorem showing that charged particles i n an arbitrary electromagnetic field with at least one ignorable spatial co ordinate remain forever tied to a given magnetic field line. Such a situati on contrasts with the significant motions normal to the magnetic field that are expected in most real three-dimensional systems. It is pointed out tha t while the significance of the theorem has not been widely appreciated, it has important consequences for a number of problems and is of particular r elevance for the acceleration of cosmic rays by shocks.