Particle capture by a slowly varying periodic potential

Particle capture by a slowly varying one-dimensional periodic potential is studied by the method of averaging [1]. For large time intervals $t\sim 1/\alpha$ ($\alpha$ is the small parameter which characterizes the rate of change of the potential), including the point of intersection of the separatrix, the solution is constructed up to the first correction terms of order relative to the leading term. The increment $\Delta I$ of the action in a complete evolution interval is also calculated in the leading order in $\alpha$.