The electron trajectories in free-electron lassers with an electromagnetic wiggler and an axially guided magnetic field are analysed with the rolativistie factor variable γ, which were usually assumed to be constant and the equations were linearized. A solution of the nonlinear relativistic equations of motion for such a system is presented. The results show that the electron trajectories are unstable, irregular, ring-shaped, corresponding to Ω0/(γ0ω)=, ≈, or <, >(1+υ30) respectively. The condition for constant γ and hence a circle projected trajectory is given.