A three-dimensional Hamiltonian formalism is used to study the particle dynamics in a free-electron laser (FEL) with a helical wiggler field and a uniform axial guide field. Oil the one hand, the construction of an original canonical transformation predicts the second constant of motion, P., 7 that permits the expansion of the Hamiltonian about a fixed point. On the other hand, the definition of the rotational variable h plays an essential role in the diagonalization of the quadratic Hamiltonian and yields two uncoupled oscillators with definite frequencies and amplitudes. Applying this variable near a fixed point brings to light Heisenberg's and the harmonic oscillator equations of motion of the particles, leading to the association of steady-state ideal helical trajectories with arbitrary trajectories. The oscillator characteristic frequencies allow one to study the different modes of propagation and to identify, and then avoid, the problematic operating conditions of the concerned FEL. The general behavior of the oscillator rotationals, (h) over cap (+/-), as functions of the normalized radius is investigated for the value 0.05 of the ratio (Ω) over cap (w)/(Ω) over cap (0). Provided that the magnitudes of the oscillator amplitudes are not too large, the electron dynamics in the neighborhood of the trajectories can be described accurately using our approach.
