Hannay’s Hoop Beyond Asymptotics

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