Rotation angles of a rotating disc as the holonomy of the Hopf fibration
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911301662408704 |
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| author | Matsumoto, Takuya |
| author_facet | Matsumoto, Takuya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04481 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rotation angles of a rotating disc as the holonomy of the Hopf fibration Matsumoto, Takuya Mathematical Physics High Energy Physics - Theory Differential Geometry Applied Physics Quantum Physics 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. |
| title | Rotation angles of a rotating disc as the holonomy of the Hopf fibration |
| topic | Mathematical Physics High Energy Physics - Theory Differential Geometry Applied Physics Quantum Physics |
| url | https://arxiv.org/abs/2512.04481 |