Hannay’s hoop beyond asymptotics

Abstract

Certain systems do not completely return to themselves when a subsystem moves through a closed circuit in physical or parameter space. A geometric phase, known classically as Hannay's angle and quantum mechanically as Berry's phase, quantifies such anholonomy. We study the classical example of a bead sliding frictionlessly on a slowly rotating hoop. We elucidate how forces in the inertial frame and pseudo-forces in the rotating frame shift the bead. We then computationally generalize the effect to arbitrary-not necessarily adiabatic-motions. We thereby extend the study of this classical geometric phase from theory to experiment via computation, as we realize the dynamics with a simple apparatus of wet ice cylinders sliding on a polished metal plate in 3D printed plastic channels.