The rotation vector fields on are , ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Group theory and symmetry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

The rotation vector fields on R3\mathbb R^3 are ξx=y∂z−z∂y\xi_x = y\partial_z - z\partial_y, ξy=z∂x−x∂z\xi_y = z\partial_x - x\partial_z, ξz=x∂y−y∂x\xi_z = x\partial_y - y\partial_x, with flows that rotate points rigidly about the xx-, yy-, zz-axes respectively; the Lie bracket satisfies [ξx,ξy]=−ξz[\xi_x,\xi_y] = -\xi_z. Take a small angle ε=0.2 rad\varepsilon = 0.2\ \mathrm{rad} and consider the four-step manoeuvre: rotate a point by angle ε\varepsilon about the xx-axis, then by angle ε\varepsilon about the yy-axis, then by angle −ε-\varepsilon about the xx-axis, then by angle −ε-\varepsilon about the yy-axis. Someone argues: "the xx-rotations cancel exactly (+ε+\varepsilon then −ε-\varepsilon), and so do the yy-rotations, so the point returns exactly to where it started; composing rotations about the xx- and yy-axes can never produce any net rotation about the zz-axis." ((a)) Explain why this argument is wrong, using the leading-order result for the net displacement of a four-step loop built from two vector fields X,YX,Y: φ−tY∘φ−tX∘φtY∘φtX≈id+t2[X,Y]\varphi^Y_{-t}\circ\varphi^X_{-t}\circ\varphi^Y_t\circ\varphi^X_t \approx \mathrm{id} + t^2[X,Y] for small tt. ((b)) Compute the leading-order net rotation angle about the zz-axis produced by the manoeuvre above, for ε=0.2 rad\varepsilon=0.2\ \mathrm{rad}.

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