High Frequency Vibrations and Non-Holonomic Mechanics

MARK LEVI
Department of Mathematics
Pennsylvania State University

Abstract:
If a pivot point of a pendulum (a rigid rod on a hinge) is forced to vibrate with high enough frequency, then the pendulum
wants to align with the direction of vibrations. In particular, if the pivot vibrates vertically, the upside-down (unstable)
position of the pendulum becomes stable. This well known but still striking phenomenon has a less well known simple
geometrical explanation which also explains with no formulas why does the Paul trap work. I will survey some recent
results on related phenomena, and will describe a perhaps unexpected connection with non-holonomic mechanics and
with the geometry of bicycle tracks.