Complex variables in classical mechanics

Question

In quantum mechanics complex numbers are absolutely essential because of the relation $$[\hat q_i,\hat p_j]=i\hbar\delta_{ij}.$$ But is complex number also essential anywhere in the formalism of classical physics, except for some convenience of mathematical calculations? Is there any example from classical physics where there is no other way out except to introduce complex variables? Is there any strong reason by which I can say that complex number are unavoidable in classical physics too?

Answer 1

In classical Hamiltonian mechanics you have the symplectic matrix $\mathbb{J}$ that has the property that $\mathbb{J}^2 = -\mathbb{I}$, and it plays a very similar role as $i$ in quantum theories. I've not really seen a situation in classical mechanics of particles where you couldn't avoid the use of the imaginary unit, but its fundamental symplectic structure (which, footnote, is structurally analogous to unitarity in many ways) does inevitably introduce $\mathbb{J}$ to phase space. But as mentioned in the comments, why would you chase around sines and cosines in a linear theory when $e^{i \omega t}$ works just as well?