In classical mechanics, are complex numbers unphysical?

Question

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Physics math without $\sqrt{-1}$

When I produce a complex final solution to a problem that began without complex coefficients at all, I have so far (with my limited expertise) tended to discard it as unphysical, unless I had to tinker some more, in which case the $i$s may square and real-ity be restored. Is this acceptable?

I understand that the Wessel plane is useful in modelling planetary motion, and many other things, but the fact (?) that you could do it all (albeit more awkwardly) without them seems to be indicative that they are a little contrived.

In short, is there a counter-example?; a physical theory (not a particular, neat, mathematical model of a physical theory) in classical mechanics that could not be formulated without complex numbers?

Answer 1

The answer is "It depends how you use complex numbers"!

If you are modelling motion on the plane, you could use $\mathbb{C}$ instead of $\mathbb{R}^{2}$ perfectly fine. In this case $1$ and $i$ are "unit vectors".

If you are working with, e.g., waves in a plane...complex numbers are quite natural.

This is markedly different from quantum theory, though.

Answer 2

No. For example, complex numbers are used in an essential way in classical electromagnetism to study networks of conductors, capacitors, and inductors.