Relativistic cyclotron motion in a polarized electric field

The relativistic cyclotron motion in a uniform magnetic field and in a transverse time-periodic electric field is studied, with particular attention to qualitative differences in the nature of the motion under linearly and circularly polarized electric fields. Theoretical analysis based on the Lagrangian formulation of the cyclotron motion allows a direct comparision for the two polarizations and confirms the earlier observation that the cyclotron motion is always integrable when the electric field is circularly polarized, but is not under a linearly polarized electric field.