Relativistic electron trajectories in free electron lasers with an axial guide field

The single-particle trajectories of relativistic electrons are studied in a magnetic field configuration which consists of a uniform axial guide field as well as a helical periodic wiggler field. The orbit equations are shown to reduce to a single nonlinear differential equation for the axial velocity, which has the form of the general elliptic equation. This equation is solved, and the classes of solutions are discussed for a wide range of experimental parameters. Solutions with constant axial velocity are found only as limiting cases of the generally anharmonic oscillations in the axial velocity.