Divergences in the iterative and perturbative methods for computing Hannay’s angle
I. Gjaja, Amitava Bhattacharjee · Physical Review A · 1990
A classical analog is obtained for Berry's nonperturbative scheme of adiabatic iteration which computes corrections to Berry's phase (or Hannay's angle) for finite values of the adiabatic parameter \ensuremath{\epsilon}. The iterative method is compared to the Lie version of adiabatic perturbation theory. Both approaches show a divergence of k!${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$, where k is the order of iteration. It is argued that the divergences are a mathematical artifact of the asymptotic methods used, not related to the physical effect of transitions, nor to the nonconservation of the action variables.