Rotation angles of a rotating disc as the holonomy of the Hopf fibration

TL;DR

This paper models a rolling disc system to reveal that its geometric rotation phase corresponds to the $U(1)$ holonomy of the Hopf fibration, linking physical motion to geometric fiber bundle properties.

Contribution

It demonstrates that the geometric phase in a rolling disc system is described by the $U(1)$ holonomy of the Hopf fibration, connecting physical rotation to geometric fiber bundle theory.

Findings

The total rotation decomposes into dynamical and geometric phases.

The geometric phase is characterized as the $U(1)$ holonomy of the Hopf fibration.

Using a Gauss map, the disc's motion is represented as a curve on $S^2$.

Abstract

This article investigates a simple kinematical model of a disc (Disc B) rolling on the edge of a fixed disc (Disc A) to study the geometric nature of rotation. The total rotation angle $\Delta$ of Disc B after one cycle is decomposed into a dynamical phase $\Delta_d$ and a geometric phase $\Delta_g$. The paper's main contribution is to demonstrate that this geometric phase can be essentially described as the $U(1)$ holonomy of the Hopf fibration with the canonical connection. By using a Gauss map to represent the disc's motion as a curve on a two-sphere ($S^2$), the work connects the physical rotation to the underlying geometry of the Hopf fiber bundle $S^3 \to S^2$ and clarifies the origin of the geometric phase.