How did the author reduce these two Euler equations?

Question

From Classical Mechanics by Taylor:

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What is this "familiar trick" that he talks about? Where did this come from? How does it work?

Answer 1

We have, from Equation $10.92$ in the book, \begin{align} \dot\omega_1 &= \Omega_b \omega_2, \\ \dot\omega_2 &= -\Omega_b \omega_1. \\ \end{align}

If we define $\eta = \omega_1 + i \omega_2,$ then \begin{align} \dot\eta &= \dot\omega_1 + i\dot\omega_2 \\ &= (\Omega_b \omega_2) + i(-\Omega_b\omega_1) & \text{substitution per Equation 10.92} \\ &= \Omega_b (\omega_2 - i\omega_1) \\ &= -i\Omega_b (i\omega_2 + \omega_1) \\ &= -i\Omega_b \eta, \end{align} as claimed in the textbook.

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That's how the "trick" works. As for where it came from, notice that the textbook's author calls this "the now familiar trick" rather than simply "the familiar trick." This strongly suggests that the author is not assuming the trick was familiar to the reader at the start of the book, but by now (having read this far) the trick should be familiar to the reader. That's why one might expect to find at least one and probably more than one example of this "trick" (or something very much like it) earlier in the book.

In any case, you don't need arcane mathematical knowledge to understand the trick; I simply performed the recommended substitution and observed how $\dot\eta = -i \Omega_b \eta$ then works out to be equivalent to Equation $10.92.$ Then I arranged the equations as shown above.

I believe this is a generally familiar trick to physicists (putting one function on the imaginary axis and one on the real axis in order to combine two real functions into one complex function), and maybe it's the sort of thing one might think to apply to a problem like this if one had a lot of prior experience doing physics; but you aren't being asked to decide to apply this trick, only to recognize that the trick works.