Advanced Classical Mechanics/Poisson Brackets

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:

$$\mathrm{d}H = \frac {\partial H}{\partial p_j}\mathrm{d}p_j + \frac {\partial H}{\partial q_j}\mathrm{d}q_j +\frac {\partial H}{\partial t}\mathrm{d}t $$

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

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 $$ H^{op}$$.