Phase transitions in the rolling of irregular cylinders and spheres

Abstract

When placed on an inclined plane, a perfect 2D disk or 3D sphere simply rolls down in a straight line under gravity. But how is the rolling affected if these shapes are irregular or random? Treating the terminal rolling speed as an order parameter, we show that there are qualitative transitions in the speed as a function of the dimension of the state space and inertia. We calculate the scaling exponents and the macroscopic lag time associated with the presence of first- and second-order transitions and describe the regimes of coexistence of stable states and the accompanying hysteresis. Experiments with rolling cylinders corroborate our theoretical results on the scaling of the lag time. Experiments with spheres reveal closed orbits and their period-doubling in the overdamped and inertial limits, respectively, providing visible manifestations of the hairy ball theorem and the doubly connected nature of S O ( 3 ) , the space of 3D rotations. Going beyond simple curiosity, our study might shed light on a number of natural and artificial systems that involve the rolling of irregular objects, ranging from nanoscale cellular transport to robotics.