The parking car has configuration space with coordinates

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

The parking car has configuration space with coordinates (x,y,θ)(x,y,\theta). Its two allowed motions are driving, D=cos⁡θ ∂x+sin⁡θ ∂yD=\cos\theta\,\partial_x+\sin\theta\,\partial_y, and turning on the spot, W=∂θW=\partial_\theta, and their bracket is the sideways field S=[W,D]=−sin⁡θ ∂x+cos⁡θ ∂yS=[W,D]=-\sin\theta\,\partial_x+\cos\theta\,\partial_y. ((a)) Compute [W,S][W,S] and [D,S][D,S]. Does bracketing again produce a direction that is not a combination of WW, DD and SS? ((b)) Show that WW, DD and SS are linearly independent at every point, so that together they span every tangent space of the configuration space. ((c)) Run the loop in the other order, starting at the origin heading along xx: drive forward ℓ=2.0 m\ell=2.0\ \mathrm{m}, turn through α=0.25 rad\alpha=0.25\ \mathrm{rad}, drive back 2.0 m2.0\ \mathrm{m}, then turn back. Predict the displacement from the bracket, then find the exact end point and compare.

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