Poisson brackets

For a mechanic system. $f(q,p,t)$ & $g(q,p,t)$ , the Possion brackets is defined as:

$$[f,g]=\frac{\partial f}{\partial p}\frac{\partial g}{\partial q}-\frac{\partial g}{\partial p}\frac{\partial f}{\partial q}$$

Notes

Properties:

1. $\frac{df}{dt}=\frac{\partial f}{\partial t}+[H,f]$
2. $[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0$
3. $[f,c]=0$
4. $[p_i,q_j]={\delta }_{ij}$