derive the Euler angles relating the axis system to the

Needs@"SpinDynamica`"D

? AxesToEuler

AxesToEuler@system1,system2,optsD gives the Euler angles 8a,b,g< by which each axis of system1 may be rotated in order to be

brought into coincide with the corresponding axis of system2, i.e. eµ2=Rz2HaLRy2HbLRz2HgL.eµ1, where mœ8x,y,z<.The equivalent result is generated by eµ2=Rz1'HgLRy1'HbLRz1HaL.eµ1, where the primed axes of system1 follow

the rotation. Each of system1 and system2 may have the form 8Z< or 8X,Z< or 8X,Y,Z<. Here Z,X and Y are

3-vectors. If only Z is specified, the two remaining axes are constructed. If X and Z is specified, the z-axis

of the constructed system is equal to Z and the x-axis is in the XZ plane. If all three axes are specified,

they must be a right-handed orthogonal set.If system2 is missing, the system 8ex,ey,ez< is assumed.

generate a randomly oriented Axis system (specifying the z-axis)

vecz = RandomReal@8−1, 1<, 83<D80.0542663, −0.371543, −0.426191<

axes = AxisSystem@veczD880.995426, 0.0627811, 0.0720152<,

80., −0.75378, 0.657127<, 80.0955388, −0.654121, −0.750332<<

derive the Euler angles relating the axis system to the original frame

8α, β, γ< = AxesToEuler@axesD81.46164, 2.41936, −1.71583<

8α, β, γ< ê °

883.7459, 138.619, −98.3097<

rotate the axis system back to the original position

Chop@RotateEuler@axes, 8α, β, γ<D881., 0, 0<, 80, 1., 0<, 80, 0, 1.<<

generate a randomly oriented Axis system (specifying two axes)

vecx = RandomReal@8−1, 1<, 83<Dvecz = RandomReal@8−1, 1<, 83<D8−0.66873, 0.962696, 0.490497<

80.67436, 0.659665, 0.101512<

axes = AxisSystem@vecx, veczD88−0.677032, 0.634817, 0.372336<,

80.190952, −0.337073, 0.921911<, 80.710749, 0.695261, 0.10699<<

derive the Euler angles relating the axis system to the original frame

8α, β, γ< = AxesToEuler@axesD81.18695, 1.4636, 2.36721<

8α, β, γ< ê °

868.0075, 83.8582, 135.631<

rotate the axis system back to the original position

Chop@RotateEuler@axes, 8α, β, γ<D881., 0, 0<, 80, 1., 0<, 80, 0, 1.<<

2 Euler.nb