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

arXiv:2512.04481v2 Announce Type: replace-cross Abstract: We address the fundamental question of how rotation angles arise in classical rigid body motion from a geometric viewpoint. As a fundamental kinematical setting, we consider a rotating-disc model. We show that the rotation angle accumulated in a cyclic motion can be naturally decomposed into a dynamical phase and a geometric phase, and that the latter is obtained from the $U(1)$ holonomy of the Hopf fibration equipped with its canonical connection through a natural weight-two representation of the fiber group. By introducing a Gauss map based on a normal vector fixed at the center of the rotating disc, the physical periodic motion is mapped to a closed curve on the two-sphere $S^2$, whose horizontal lift in the Hopf bundle $S^3 \to S^2$ determines the geometric phase through the weight-two representation. This framework provides a conceptually clear interpretation of rotation angles as geometric phases and clarifies the geometric origin of their decomposition.

Abstract:We address the fundamental question of how rotation angles arise in classical rigid body motion from a geometric viewpoint. As a fundamental kinematical setting, we consider a rotating-disc model. We show that the rotation angle accumulated in a cyclic motion can be naturally decomposed into a dynamical phase and a geometric phase, and that the latter is obtained from the $U(1)$ holonomy of the Hopf fibration equipped with its canonical connection through a natural weight-two representation of the fiber group. By introducing a Gauss map based on a normal vector fixed at the center of the rotating disc, the physical periodic motion is mapped to a closed curve on the two-sphere $S^2$, whose horizontal lift in the Hopf bundle $S^3 \to S^2$ determines the geometric phase through the weight-two representation. This framework provides a conceptually clear interpretation of rotation angles as geometric phases and clarifies the geometric origin of their decomposition.

36 pages, 5 figures. Revised version with substantial improvements and clarifications, in particular concerning the relation between the geometric phase and the holonomy of the Hopf fibration

Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Differential Geometry (math.DG); Applied Physics (physics.app-ph); Quantum Physics (quant-ph)

arXiv:2512.04481 [math-ph]

(or arXiv:2512.04481v2 [math-ph] for this version)

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

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