旋转圆盘的旋转角度作为霍普夫纤维化的holonomy
Rotation angles of a rotating disc as the holonomy of the Hopf fibration
AI总结：
本文通过研究圆盘滚动模型，揭示了旋转角度与霍普夫纤维化holonomy之间的几何联系。
摘要：
本文研究了一个简单的动力学模型，即一个固定圆盘（圆盘A）边缘上滚动的圆盘（圆盘B），以研究旋转的几何本质。圆盘B经过一个周期后的总旋转角度Δ被分解为一个动力学相位Δ_d和一个几何相位Δ_ g。本文的主要贡献是证明这个几何相位本质上可以描述为霍普夫纤维化的标准连接的U(1) holonomy。通过使用高斯映射将圆盘的运动表示为二维球面（S²）上的曲线，本文将物理旋转与霍普夫纤维丛S³→S²的底层几何联系起来，并阐明了几何相位的起源。
英文摘要：
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.
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