I'm reading LittleJohn's notes on Rotations in Ordinary Space on Quantum Mechanics. Link: http://bohr.physics.berkeley.edu/classes/221/1011/notes/classrot.pdf. I'm trying the last question given in the document one the last page:
It is claimed that every proper rotation can be written in Euler angle form. Find the Euler angles $(\alpha, \beta, \gamma)$ for the rotation $R(\hat{x}, π/2)$.
The formula for the Euler Rotation being used is given by $(58)$ in the notes.
My question is: is there a systematic way to figure out the Euler Angles, via the use of some formula, for relatively easy rotations like this? Or would have to figure out how the basis vectors transform under the said rotation and then try and figure out the Euler Angles by inspection or solving a system of equations by representing the sequence of Euler rotations in matrix form (if that's possible)?
Also, how does one go about solving this problem for rotations about an arbitrary axis? Is that a very difficult problem?
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