Dynamics of a charged particle in a circularly polarized traveling electromagnetic wave

The relativistic motion of a charged particle in a transverse circularly or almost circularly polarized homogeneous electromagnetic wave is studied using the Hamiltonian formalism. First, the case of a circularly and almost circularly polarized traveling wave propagating in a nonmagnetized space is studied. In the case of an almost circularly polarized wave, it is shown that the charged particle has an average velocity along the propagation direction of the wave. The same result is derived considering a cold electron plasma. The case of a traveling wave propagating along a constant homogeneous magnetic field is then considered. Using canonical transformations, it is shown that the equations of motion can be derived from an autonomous Hamiltonian which has two degrees of freedom and a first integral. As a consequence, the system is completely integrable. An equation is found for the particle energy when it is initially resonant. This equation is solved exactly, and the asymptotic solution is obtained. The expression for the energy allows a solution for the system in terms of quadratures, and in consequence the asymptotic solution for all the variables. The case of an almost circularly polarized wave propagating along a constant homogeneous magnetic field is also studied. Finally, a magnetic field gradient is considered, and new acceleration mechanisms are found.