Advanced Classical Mechanics/Poisson Brackets

Subject classification: this is a mathematics resource.

Poisson Brackets

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

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

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

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

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

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

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

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

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

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

References

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Advanced Classical Mechanics

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