傅科摆摆平面转动的平行移动解释
EXPLANATION OF THE FOUCAULT PENDULUM BASED ON THE PARALLEL TRANSPORT OF VECTORS
吴葛一1
作者信息
- 1. 清华大学钱学森力学班,北京 100084
- 折叠
摘要
傅科摆摆平面的转动可以通过牛顿力学或矢量的平行移动解释,但后者要求深入理解微分几何,对力学读者不够友好.本文从几何直观出发,解释曲面上矢量平行移动的数学定义,建立起平行移动与纯滚动的关联.再用动量定理证明摆锤整摆动周期时刻的速度矢量沿球面上的纬线圆平行移动.最后通过图像解释、数学计算两种方法,求解摆平面的转动角速度.读者只需了解矢量运算和动量定理,即可深入理解傅科摆,建立力学和几何的关联.
Abstract
The precession of the swing plane of the Foucault pendulum can be explained using either Newtonian mechanics or the parallel transport of vectors.The latter one,however,requires a profound understanding of differential geometry,making it less accessible to readers with a background in mechanics.This paper offers a geometrically intuitive explanation of the parallel transport of vectors on a curved surface,establishing its correlation with pure rolling.By applying the principle of momentum,it is proven that the velocity vector at intervals of the oscillation period of the pendulum undergoes parallel transport along a latitude on the sphere.The angular velocity of the swing plane precession is then determined through both graphical interpretation and mathematical calculation.This paper only requires readers to possess a basic understanding of vector operations and the principle of momentum,enabling them to have a deeper comprehension of the Foucault pendulum and establish the connection between mechanics and geometry.
关键词
傅科摆/微分几何/矢量平行移动/纯滚动Key words
the Foucault pendulum/differential geometry/parallel transport of vectors/pure rolling引用本文复制引用
出版年
2024
NETL
NSTL
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