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 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. 
Frank C. Jones, J. Randy Jokipii and Matthew G. Baring