A satellite gimbal is rotated (actively) by about the fixed -axis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Lie groups and Lie algebras: rotations, SU(2) and the double cover

Problem

A satellite gimbal is rotated (actively) by 90∘90^\circ about the fixed xx-axis, then by 90∘90^\circ about the fixed yy-axis; in the convention ghgh means hh first, so the combined motion is Ry(90∘)Rx(90∘)\mathsf R_y(90^\circ)\mathsf R_x(90^\circ). An engineer argues: 'each rotation is generated by JxJ_x and JyJ_y, so the combined motion is simply exp⁡ ⁣[−iπ2(Jx+Jy)]\exp\!\big[-i\tfrac{\pi}{2}(J_x+J_y)\big], a single rotation about the axis (1,1,0)/2(1,1,0)/\sqrt2.' ((a)) Compute the true combined rotation matrix Ry(90∘)Rx(90∘)\mathsf R_y(90^\circ)\mathsf R_x(90^\circ) and find where it sends e^x=(1,0,0)\hat{\mathbf e}_x=(1,0,0). ((b)) Compute where exp⁡ ⁣[−iπ2(Jx+Jy)]\exp\!\big[-i\tfrac{\pi}{2}(J_x+J_y)\big] sends e^x\hat{\mathbf e}_x and compare with part (a). ((c)) Explain why the engineer's shortcut fails, and say whether the SU(2)SU(2) element representing the true combined rotation is unique.

Related theory

Browse the problem library