Rotation angles of a rotating disc -- A toy model exhibiting the geometric phase --

Abstract

In this paper, we consider a simple kinematic model, which is a rotating disc on the edge of another fixed disc without slipping, and study the rotation angle of the rotating disc. The rotation angle consists of two parts, the dynamical phase $\Delta_d$ and the geometric phase $\Delta_g$. The former is a dynamical rotation of the disc itself, and the geometric motion of the disc characterizes the latter. In fact, $\Delta_g$ is regarded as the geometric phase appearing in several important contexts in physics. The clue to finding the explicit form of $\Delta_g$ is the Baumkuchen lemma, which we called. Due to the Gauss-Bonnet theorem, in the case that the rotating disc comes back to the initial position, $\Delta_g$ is interpreted as the signed area of a two-sphere enclosed by the trajectory of the Gauss vector, which is a unit normal vector on the moving disc. We also comment on typical models sharing the common underlying structure, which include Foucault's pendulum, Dirac's monopole potentials, and Berry phase. Hence, our model is a very simple but distinguished one in the sense that it embodies the essential concepts in differential geometry and theoretical physics such as the Gauss-Bonnet theorem, the geometric phase, and the fiber bundles.