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?

  • $\begingroup$ I'm maybe confused. May you write the Cauchy Riemann equations to see if we agree, I wouldn't know what in CR equations plays the role of the time $t$. $\endgroup$ – Mauricio Sep 5 '18 at 16:10
  • $\begingroup$ These are different from the Cauchy-Riemann equations. In the CR equations, going from the first equation to the second equation swaps the denominator on one side with the denominator on the other side. In Hamilton's equations, going from the first to the second equation swaps the numerator on one side with the denominator on the other. $\endgroup$ – probably_someone Sep 5 '18 at 16:29

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.


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.


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