Gaussian curvature and gyroscopes
Abstract
We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk which is equal to the magnetic force on a point charge moving in a magnetic field normal to the surface, of magnitude equal to the Gaussian curvature, and of charge equal to the disk's axial spin. In a special case, this demonstrates that the precession of Lagrange's top is due to the curvature of a sphere determined by the parameters of the top.
Cite
@article{arxiv.1607.03217,
title = {Gaussian curvature and gyroscopes},
author = {Graham Cox and Mark Levi},
journal= {arXiv preprint arXiv:1607.03217},
year = {2017}
}
Comments
12 pages, 10 figures; v2: minor changes to notation and exposition, Section 2.3 and Figure 4 added