New analysis, simulations, and 3D printing expand the scope and appeal of a classical geometric phase.
Falling cats and Olympic divers share the ability to twist, spin, and reorient themselves to land on their feet or make minimal splash. To accomplish that feat, they bend and contort to make a complete loop in their body’s “shape space,” so their bodies end up in the same shape they started in. But in physical space, they don’t end up where they started: They rotate through a finite angle. Michael Berry described that acquired rotation, a so-called geometric phase, 35 years ago for quantum systems, and the phase now bears his name. Shortly thereafter, John Hannay extended the concept to classical analogues, for which the iconic example is a bead sliding frictionlessly on a horizontal, rotating, noncircular hoop. After one rotation, the hoop returns to its starting orientation, but the bead will have moved by an angle that depends only on the hoop geometry, not on its rotation speed.
Hannay’s original analysis considered only slow, adiabatic rotations. John Lindner and colleagues at The College of Wooster now generalize it to arbitrary motions, including regimes that can be readily realized in laboratory experiments. In the hoop’s rotating frame of reference, the bead experiences multiple fictitious forces, of which the Coriolis and centrifugal forces are the more familiar. But it is the Euler pseudo force, arising from the hoop’s initial angular acceleration, that sets the bead in motion in the hoop frame. Numerical simulations (see the figure) showed that the resulting trajectory in the lab frame (blue curve, right) can be quite complex and include multiple kinks. The researchers validated their simulations experimentally, with a cylinder of wet ice, weighted down by a steel ball, that slid around an elliptical track fashioned from 3D-printed plastic rails affixed to an aluminum sheet on a turntable. Converting a classic, idealized example into a realized experiment offers a firsthand opportunity to explore geometric phases in all their classical and quantum forms. (H. Bae, N. Ali, J. F. Lindner, Chaos 28, 083107, 2018.)