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 in an arbitrary electromagnetic field with at least one ignorable spatial coordinate remain forever tied to a given magnetic field line. Such a situation contrasts with the significant motions normal to the magnetic field that are expected in most real three-dimensional systems. It is pointed out that while the significance of the theorem has not been widely appreciated, it has important consequences for a number of problems and is of particular relevance for the acceleration of cosmic rays by shocks.