The geometric phase and the dry friction of sleeping tops on inclined planes

Sven Barthmann,Thomas M Fischer

doi:10.1088/2399-6528/ac1874

Abstract

We report on the motion of a spinning sleeping top on an inclined plane. Below a critical inclination angle the sleeping tops are force free. The trajectory of a sleeping top on weakly inclined planes in the adiabatic limit is invariant of the angular frequency of the top and thus invariant under a rescaling of the time, however not invariant under time reversal. The stationary trajectory of the sleeping top is characterized by its Hannay type geometric angle to the in plane horizontal direction. At larger inclinations of the plane the stationary motion of the top becomes unstable and the top accelerates downhill. The behavior points towards a complex law of dry friction of the contact point between the top tip and the material of the inclined plane that depends on a slip parameter. We propose a phenomenological law of dry friction that can explain the relaxation of the top into the sleeping position, the geometric behavior of the top trajectories, and the instability of the stationary motion at larger inclination angles.

Highlights

A geometric phase is a phase difference acquired over each period of a cyclic adiabatic process
The geometric phase is accumulated under adiabatic, i.e. slow enough conditions independent of the speed, while the system follows a path in parameter space
In the adiabatic limit a sleeping spinning metal top moves on a weakly inclined plane along a trajectory that is invariant under rescaling of time, but not invariant under time reversal

Summary

INTRODUCTION

A geometric phase is a phase difference acquired over each period of a cyclic adiabatic process. The geometric phase named Berry-phase in quantum sytems and Hannay-angle in classical systems is invariant under rescaling of the time schedule with which one passes the path in parameter space It carries along a gauge freedom of choice of reference points and the concept has been used in high energy particle physics [4], in solid state physics of topological materials [5,6,7], in the explanation of the rotation of the Foucaultpendulum[8], in the explanation of the propulsion of active swimmers in low Reynolds number fluids[9, 10], the propulsion of light in twisted fibers[11, 12], the propulsion of acoustic [13] or stochastic[14] waves, the motion of edge waves in coupled gyroscope lattices[15], the rolling of nucleons[16] and in the control of the transport of macroscopic[17] and colloidal[18,19,20,21] particles above magnetic lattices. The velocity independent friction law of Coulomb is clearly violated and we must replace it by a dry dynamic friction force law that depends on the sliding velocity

EXPERIMENTS

PHEMOMENOLOGICAL MODEL

CONCLUSIONS