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

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.