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:

$\{f,g\}=\sum _{i=1}^N(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}).$

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:

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

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

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

$\frac{d}{dt}⟨A^{op}⟩=\frac{1}{i\hslash }⟨[A^{op},H^{op}]⟩+⟨\frac{\partial A^{op}}{\partial t}⟩,$

where $A^{op}$ is any operator of quantum mechanics, $⟨A^{op}⟩$ 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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