I would like to see if this idea has any applications:


So CR equations are given by:


$$ \frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} ; \ \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$


And Hamilton equations are given by:


$$ \dot{p}=-\frac{\partial H}{\partial q} ; \ \dot{q}= \frac{\partial H}{\partial p}$$


Now these conjugate equations ask for an analogy between them, so I tried the next thing:


I am trying to find $u(p,q),v(p,q)$ s.t:


$$\dot{p}=-\frac{\partial H}{\partial q}=\frac{\partial u}{\partial p}=-\frac{\partial v}{\partial q}$$
$$\dot{q}= \frac{\partial H}{\partial p} = \frac{\partial u}{\partial q} =\frac{\partial v}{\partial p}$$


So we can see that we can find a complex-analytic representation of the hamiltonian, $H$:


$$f(p,q)=u(p,q) + \imath (H(p,q)+c)$$


The only restriction is for $u(p,q)$ which by integration we can find it.


Does this represnetation has any applications in mathematical physics or complex analysis? I guess it's already known.


Thanks in advance, Alan.



Similarity between Cauchy-Riemann eqs and Hamilton equations.
Doesn't this only work if $H(p,q)$ is a harmonic function in the plane?


If that's the case, we have 


$\{u,H\}=\frac{\partial H}{\partial p}\frac{\partial u}{\partial q}-\frac{\partial H}{\partial q}\frac{\partial u}{\partial p}=|\nabla H|^2=-|\nabla u|^2$


where $\{\cdot,\cdot\}$ is the Poisson bracket. Since $\frac{du}{dt}=\{u,H\}$ this tells us how fast $u$ is changing.


