 Poisson bracket in complex canonical coordinates

ID: poisson-bracket-in-complex-canonical-coordinates

For

z_{j} = q_{j} + i p_{j}

and

\overset{z}{ˉ}_{j} = q_{j} - i p_{j}

{f, g} = - 2 i \frac{\partial f}{\partial z _{j}} \frac{\partial g}{\partial z ˉ _{j}} + 2 i \frac{\partial g}{\partial z _{j}} \frac{\partial f}{\partial z ˉ _{j}} .

This follows from

\partial_{q_{j}} = \partial_{z_{j}} + \partial_{\overset{z}{ˉ}_{j}}

and

\partial_{p_{j}} = i \partial_{z_{j}} - i \partial_{\overset{z}{ˉ}_{j}}

 New to topics? Read the docs here!