Poisson bracket

$\{f,g\}:={\sum }_{i=i}^n\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$ is the Poisson bracket expressed using canonical coordinates. It is

• anti-commutative : $\{f,g\}=-\{g,f\}$
• bilinear : $\{af+bg,h\}=a\{f,h\}+b\{g,h\}$
• follows the Leibniz rule : $\{fg,h\}=\{f,h\}g+f\{g,h\}$
• follows the Jacobi identity: $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$

In the above we had

• $p,q\in {\mathbb{R}}^n$ as coordinates in the Phase Space
• $f,g,h:{\mathbb{R}}^{2n+1}\to \mathbb{R}$ as real valued scalar fields, with inputs $p,q\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$

Intuition

( based on this quora answer)

For a Force $F$, and by the definition of Hamilton’s Equations of Motion, we have:

$$\begin{array}{rll}\dot{F}(p_i,q_i)& =\frac{\partial F}{\partial q_i}{\dot{q}}_i+\frac{\partial F}{\partial p_i}{\dot{p}}_i& \text{Using Hamilton’s Equations of Motion}\\ & =\frac{\partial F}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial H}{\partial q_i}& \text{By the definition of the Poisson Bracket}\\ & =\{F,H\}& \end{array}$$

This makes the Poisson bracket an analytically simple way of understanding and working with Forces. Most of the results applicable to Poisson brackets extend to Commutator results in quantum theory.