Poisson Brackets

The Poisson bracket of any two functions, ${\displaystyle f(p_{i},q_{i},t)}$ and ${\displaystyle g(p_{i},q_{i},t)}$, is:

${\displaystyle \{f,g\}=\sum _{i=1}^{N}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right).}$

In two dimensions, the multivariable chain rule, is ${\displaystyle \mathrm {d} f(x,y)={\frac {\partial f}{\partial x}}\mathrm {d} x+{\frac {\partial f}{\partial y}}\mathrm {d} y}$. Using implied summation notation (for the index j), we apply this to Hamilton's equations:

${\displaystyle \mathrm {d} H={\frac {\partial H}{\partial p_{j}}}\mathrm {d} p_{j}+{\frac {\partial H}{\partial q_{j}}}\mathrm {d} q_{j}+{\frac {\partial H}{\partial t}}\mathrm {d} t}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}f(p,q,t)&={\frac {\partial f}{\partial q}}{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {\partial f}{\partial p}}{\frac {\mathrm {d} p}{\mathrm {d} t}}+{\frac {\partial f}{\partial t}}\\&={\frac {\partial f}{\partial q}}{\frac {\partial H}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial H}{\partial q}}+{\frac {\partial f}{\partial t}}\\&=\{f,H\}+{\frac {\partial f}{\partial t}}\,.\end{aligned}}}$

As an aside, we note a connection to Quantum Mechanics: Ehrenfest theorem involves the operators and expectation values of quantum mechanics. It states: ^[1]

${\displaystyle {\frac {d}{dt}}\langle A^{op}\rangle ={\frac {1}{i\hbar }}\langle [A^{op},H^{op}]\rangle +\left\langle {\frac {\partial A^{op}}{\partial t}}\right\rangle ~,}$

where ${\displaystyle A^{op}}$ is any operator of quantum mechanics, ${\displaystyle \langle A^{op}\rangle }$ is its expectation value, and

${\displaystyle [A^{op},H^{op}]=A^{op}H^{op}-H^{op}A^{op}}$

is the commutator of ${\displaystyle A^{op}}$ and ${\displaystyle H^{op}}$.

References

1. ↑ Smith, Henrik (1991). Introduction to Quantum Mechanics. World Scientific Pub Co Inc. pp. 108–109. ISBN 978-9810204754.