Wednesday 9 January 2019

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.





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