We study electron motion in electromagnetic (EM) fields in the radiation-dominated regime. It is shown that the electron trajectories become close to some asymptotic trajectories in the strong field limit. The description of the electron dynamics by this asymptotic trajectories significantly differs from the ponderomotive description that is barely applicable in the radiation-dominated regime. The particle velocity on the asymptotic trajectory is completely determined by the local and instant EM field. The general properties of the asymptotic trajectories are discussed. In most of standing EM waves (including identical tightly-focused counter-propagating beams) the asymptotic trajectories are periodic with the period of the wave field. Furthermore, for a certain model of the laser beam we show that the asymptotic trajectories are periodic in the reference frame moving along the beam with its group velocity that may explain the effect of the radiation-reaction trapping.
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