A flywheel spins freely on a fixed axle

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 flywheel spins freely on a fixed axle. Its configuration is the angular position

\theta

of a marked point on its rim, with

\theta

and

\theta+2\pi

identified, so the configuration space is the circle

S^1

, drawn as the unit circle

x^2+y^2=1

with

x=\cos\theta

y=\sin\theta

. ((a)) Chart

A

projects each point of the circle from the point

\theta=\pi

onto the

y

-axis:

u=\dfrac{y}{1+x}

. Show that

u=\tan(\theta/2)

, that

A

is defined and smooth for every

\theta\ne\pi

, and that it is a bijection of that domain onto

\mathbb{R}

. ((b)) Chart

B

projects from the point

\theta=0

u'=\dfrac{y}{1-x}

. Show that

u'=\cot(\theta/2)

, defined and smooth for every

\theta\ne0

. ((c)) On the overlap of the two domains, find

u'

as a smooth function of

u

, and confirm it has no singularity there. ((d)) Explain why

\{A,B\}

is an atlas that makes the flywheel's configuration space a smooth one-dimensional manifold, and why chart

A

alone could never do the job.

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