# Intrinsic $Z$-$Y$-$Z$ Euler angle sequence

Consider Euler angles $$\alpha$$, $$\beta$$, $$\gamma$$ with the intrinsic $$Z$$-$$Y$$-$$Z$$ convention. The effective rotation $$S(\alpha, \beta, \gamma)$$ can be written as: $$$$S(\alpha, \beta, \gamma) = S_z(\gamma)S_y(\beta)S_z(\alpha).$$$$ $$S_i(\theta)$$ denotes the rotation about $$i$$ axis by an angle $$\theta$$, where $$i \in \{x, y, z\}$$. According to my understanding, $$$$S_y(\beta) = \pmatrix{\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta }.$$$$ However, "Mathematical Methods for Physicists" by George Arfken, Hans Weber and Harris uses $$$$S_y(\beta) = \pmatrix{\cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta }$$$$ for calculating the effective rotation $$S(\alpha, \beta, \gamma)$$, stating the reason as "the signs of $$\sin \beta$$ have to be consistent with a cyclic permutation of the axis numbering". Could someone explain this inconsistency in the signs?

Edit: The matrices used for $$S_z(\alpha)$$ and $$S_z(\gamma)$$ in the text are, \begin{align} S_z(\alpha) &= \pmatrix{\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 } \\ S_z(\gamma) &= \pmatrix{\cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 }. \end{align}

• @CosmasZachos It is clear, thank you! Commented Jan 27, 2022 at 5:05

Your "according to my understanding" rotation around $$\hat y$$, (x goes towards z when you draw the diagram, as you should), is unlike the other two, so you paved your way to dyslexia purgatory.
Normally, one chooses a consistent and memorable scheme where all three axes’ rotations are on an equal footing, either all clockwise, or all anti-clockwise, $$(x,y,z)\to (y,z,x)\to (z,x,y), ~~\leadsto \\ \hat x \times \hat y = \hat z ,\\ \hat y \times \hat z = \hat x ,\\ \hat z \times \hat x = \hat y ,$$ and all three rotations are anti-clockwise around their respective axes.
(This corresponds to an 120 degree rotation, anti clockwise, around the axis $$\hat x + \hat y+ \hat z$$.)