Two independent electric motors each turn their own shaft freely

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

Two independent electric motors each turn their own shaft freely, with no gearing between them. Shaft

1

's angular position is

\theta_1

and shaft

2

's is

\theta_2

, each identified mod

2\pi

, so the configuration space of the pair is the torus

T^2=S^1\times S^1

. ((a)) Using the circle charts

u_i=\tan(\theta_i/2)

(excluding

\theta_i=\pi

) and

u_i'=\cot(\theta_i/2)

(excluding

\theta_i=0

) for

i=1,2

, list the four charts obtained by choosing, independently for each shaft, the unprimed or primed coordinate, and state which values of

(\theta_1,\theta_2)

each chart excludes. ((b)) Write down the transition map from the chart

(u_1,u_2)

to the chart

(u_1',u_2')

on their common domain. ((c)) Compute its Jacobian determinant and show it is never zero there. ((d)) Explain why four charts, and not fewer, are needed to cover the torus by this construction.

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