Two-timescale analysis of accelerated planar motions

Two-timescale analysis has long been known as a particularly useful tool for studying dynamical systems with two different length scales. The parameter space of some accelerated worldlines shows such a situation. In this paper we find different sets of solutions for the equations of motion governing these worldlines by using two-timescale expansions. The general properties of these solutions are examined. This helps to derive simple schemes that reproduce secular effects of interacting forces. Rigorous prescriptions are given for computing the leading order motion which highlight the importance of dissipative terms in the force functions. In the end, we discuss some potential ambiguities in the two-timescale approach and possible deviations these prescriptions may have, from the exact worldlines.