PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry
Invariance, Ch Manifolds, charts and tangent spaces
Problem
A small bead of negligible size is threaded, without friction, on a rigid circular hoop of radius a. The hoop is mounted on a vertical diameter and is spun about that diameter by a motor at a prescribed, externally imposed rate Ω (the motor's speed is set by the experimenter and does not depend on where the bead is). The bead's position on the hoop is given by the angle ϕ, measured in the plane of the hoop from the point where the hoop crosses the vertical axis, with ϕ and ϕ+2π identified.
((a)) Explain why the hoop's own rotation angle is not a coordinate of the bead's configuration space, and identify that configuration space as a manifold, stating its dimension.
((b)) A single chart ϕ∈[0,2π) is proposed to describe every position of the bead. Say precisely where this chart fails to be continuous, and exhibit two charts, each a projection of the sort used for the circle, that together form a smooth atlas.
((c)) Give the transition map between your two charts and confirm it is smooth on their overlap.