Poisson Bracket

For two arbitrary functions $f$ and $g$, the Poisson bracket (with respect to the conjugate variables $\mathbf{q}$ and $\mathbf{p}$) is defined as ${f, g}{q,p} = \sum{k=1}^{N} \left (\frac{\partial f}{\partial q_k} \frac{\partial g}{\partial p_k} - \frac{\partial f}{\partial p_k} \frac{\partial g}{\partial q_k} \right)$¹.