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?
A: I think you've mistaken something along the way. The Cauchy-Riemann equations relate to the partial derivatives of two functions of two variables, say $u(x,y)$ and $v(x,y)$.  It provides a relationship between the partials of those functions with respect to each of the two independent variables.
The Hamiltonian equations are functions of (at least) 3 variables, some of which are dependent, i.e. $p(t)$ and $q(t)$, and one of which is independent, namely $t$.  The partials in that case relate changes in the dependent variables to partials of the Hamiltonian with respect to the dependent variables.
So that's different.
A: *

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


*For the canonical Poisson structure
$$ \{z, z\} ~=~0, \qquad \{\bar{z},z\}~=~i, \qquad \{\bar{z}, \bar{z}\} ~=~0, \tag{1}$$
in complex coordinates
$$ z ~=~\frac{q+ip}{\sqrt{2}},\tag{2} $$
the holomorphic condition reads
$$ 0~=~\bar{\partial} f ~=~i\{z,f\},\tag{3} $$
which is equivalent to the Cauchy-Riemann equations.


*The Hamilton's equations in complex coordinates read
$$ \dot{z}~=~\{ z, H\}~=~-i\bar{\partial}H.\tag{4},$$
cf. e.g. my Phys.SE answer here.
