The configuration space of a double pendulum is the torus with angular coordinates

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

The configuration space of a double pendulum is the torus

T^2

with angular coordinates

(\theta_1, \theta_2)

, each defined modulo

2\pi

. Near a given configuration, choose real values of

\theta_1

and

\theta_2

that vary continuously. On that neighbourhood,

a = \theta_1 + \theta_2

and

b = \theta_1 - \theta_2

are new local coordinates. At some instant the pendulum's angular velocity is the tangent vector

v = 0.5\,\partial_{\theta_1} - 0.2\,\partial_{\theta_2}

(units

\mathrm{rad\,s^{-1}}

). Find the components

v^a

v^b

v

in the

(a,b)

coordinates.

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