Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Takuya Matsumoto

TL;DR

The paper models a disc rolling on the edge of a fixed disc to study geometric phase, showing the total rotation decomposes into a dynamical part and a geometric part. By mapping the disc's orientation to a Gauss curve on $S^2$ and identifying the geometric phase with the $U(1)$ holonomy of the Hopf fibration $S^3\to S^2$, it provides a deep geometric interpretation of rotation via the canonical connection and horizontal lifts. The work connects kinematics to fiber-bundle geometry, expressing the geometric phase both as a line integral on the Gauss curve and as the holonomy of the Hopf connection, while clarifying the role of covering spaces in tracking the real-valued geometric phase. This offers a unified geometric framework with implications for understanding Foucault-like pendula and other geometric-phase phenomena in simple mechanical systems.

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 $Δ$ of Disc B after one cycle is decomposed into a dynamical phase $Δ_d$ and a geometric phase $Δ_g$. 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.

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

TL;DR

The paper models a disc rolling on the edge of a fixed disc to study geometric phase, showing the total rotation decomposes into a dynamical part and a geometric part. By mapping the disc's orientation to a Gauss curve on and identifying the geometric phase with the holonomy of the Hopf fibration , it provides a deep geometric interpretation of rotation via the canonical connection and horizontal lifts. The work connects kinematics to fiber-bundle geometry, expressing the geometric phase both as a line integral on the Gauss curve and as the holonomy of the Hopf connection, while clarifying the role of covering spaces in tracking the real-valued geometric phase. This offers a unified geometric framework with implications for understanding Foucault-like pendula and other geometric-phase phenomena in simple mechanical systems.

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 of Disc B after one cycle is decomposed into a dynamical phase and a geometric phase . The paper's main contribution is to demonstrate that this geometric phase can be essentially described as the 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 (), the work connects the physical rotation to the underlying geometry of the Hopf fiber bundle and clarifies the origin of the geometric phase.

Paper Structure

This paper contains 16 sections, 12 theorems, 128 equations, 4 figures.

Table of Contents

  1. Introduction and summary
  2. The physical model
  3. The central question and decomposition of rotation angle
  4. Geometric interpretation via Gauss map
  5. The Hopf fibration and holonomy
  6. Contents
  7. The model: Rotation angles of a rotating disc
  8. Hopf fibration
  9. Definition of the Hopf fibration
  10. Canonical connection
  11. Horizontal and vertical subspaces
  12. Horizontal lift and parallel displacement
  13. Rotation angle as the holonomy
  14. Geometric phase and $U(1)$ holonomy
  15. Geometric phase as the coordinate of $U(1)$ fiber
  16. ...and 1 more sections

Key Result

Proposition 2.5

The geometric phase for the motion eq:motion is given by the limit of the line integral along the regularized curve $\gamma(\epsilon)$ in eq:ga-reg ,

Figures (4)

  • Figure 1: Disc A is fixed on $xy$-plane and disc B rolls on the edge of the disc A without slipping.
  • Figure 2: The Gauss map $\bm{g}$ from the model (a) to two-sphere (b)
  • Figure 3: The curve $\gamma\subset S^2$ is the orbit of the Gauss vector $\bm{g}(t)$ whose orientation is induced by the parameter $t$ . $A_+ (A_-)$ is the area enclosed by $\gamma$ on the left (right) side of the figure. The topological index $I_+ (I_-)$ is the number of poles at $(0,0,\pm1)\in S^2$ contained in $\gamma$ on the left (right) side. The above picture corresponds to the case $I_+=I_-=1$ .
  • Figure 4: The moving frames $\{\bm{g}'(s)\,, \bm{\nu}(s)\}$ and $\{\bm{e}_1\,,\bm{e}_2\}$ at $\bm{g}(s)\in S^2$ . The vectors $\bm{e}_1$ and $\bm{e}_2$ are oriented to east and north, respectively. $\phi$ is the angle between the local frames.

Theorems & Definitions (34)

  • Remark 2.1
  • Definition 2.3: Gauss map
  • Definition 2.4: Regularized Gauss curve
  • Proposition 2.5: MTY
  • Theorem 2.6: Theorem 3.15 in MTY
  • Definition 3.1: Hopf fibration
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.5: Fundamental vector field
  • ...and 24 more