hamiltonian formalism - Relationship between complex analysis and Hamilton's canonical equation

I recently came across a mathematical field called complex analysis.

There was an important equation called Cauchy-Riemann equation.

When I saw it at first, I recalled a book's sentence stating that

A: Why did we substitute the Newtonian equations with that efforts?

B: It's because it is beautiful in terms of mathematics.

Hamilton's canonical equation is good because it is beautiful (in terms of mathematics)

From a Quantum mechanics book.

It was

$$\begin{align} \frac {dq}{dt}& =\frac {\partial H}{\partial p} \\ -\frac {dp}{dt}& = \frac{\partial H}{\partial q}. \end{align}$$

Doesn't this look similar to the Cauchy-Riemann equation? (Integrating $u$ makes $v$ and vice versa.)

I guess the reason is of something like the CR equation.

So my question is:

Is there any relationship between complex analysis and Hamilton's canonical equation?

Answer

Comments to the post (v5):

1. 
   

   Complex structures and symplectic structures can co-exist, notably for Kähler manifolds.

   
   


2. 
   

   For the canonical Poisson structure

   $$\{z, z\} ~=~0, \qquad \{\bar{z},z\}~=~i, \qquad \{\bar{z}, \bar{z}\} ~=~0,$$ in complex coordinates $$z ~=~\frac{q+ip}{\sqrt{2}},$$ the holomorphic condition read $$0~=~\bar{\partial} f ~=~i\{z,f\},$$ which is equivalent to the Cauchy-Riemann equations.