# Definition The Poisson bracket is defined by: $ \{f, g\} \equiv \sum_{i} \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}}-\frac{\partial g}{\partial q_{i}} \frac{\partial f}{\partial p_{i}}, \quad f, g \in \mathcal{C}(P) $ where $\mathcal{C}(P)$ is the [[vector space]] of [[Observable (Classical Mechanics)|observables]]. It forms a [[Lie Algebra|Lie bracket]] on $\mathcal{C}(p)$, is antisymmetric, and obeys the [[Jacobi Identity]]. # In Term of the Poisson Structure Writing $\pmb{\text{d}}F = (\frac{\partial F}{\partial q^i}, \frac{\partial F}{\partial p_i})$, and denoting the [[Poisson Structure|Poisson Form]] by $B$, then we have: $ \{F,G\} = B(\pmb{\text{d}}F, \pmb{\text{d}}G) = \pmb{\text{d}}F \cdot \mathbb{J} \nabla G $ here, $\mathbb{J}$ is the [[Symplectic Matrix]] and $\nabla G$ is the *naive gradient* of $G$, i.e., the row vector $\pmb{\text{d}}G$ interpreted as a column vector. It is straightforward to verify this works. For example, assume that $n = 2$, we end up with: $ \begin{align} \begin{pmatrix} \frac{\partial F}{\partial q^1} \frac{\partial F}{\partial q^2} \frac{\partial F}{\partial p_1} \frac{\partial F}{\partial p_2} \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{\partial G}{\partial q^1}\\ \frac{\partial G}{\partial q^2}\\ \frac{\partial G}{\partial p_1}\\ \frac{\partial G}{\partial p_2} \end{pmatrix} &= \left(\frac{\partial F}{\partial q^1}\frac{\partial G}{\partial p_1} - \frac{\partial G}{\partial q^1}\frac{\partial F}{\partial p_1}\right) + \left(\frac{\partial F}{\partial q^2}\frac{\partial G}{\partial p_2} - \frac{\partial G}{\partial q^2}\frac{\partial F}{\partial p_2}\right)\\ &= \{F, G\} \end{align} $