Editing Poisson algebra (section)

===Symplectic manifolds===
The space of real-valued [[smooth function]]s over a [[symplectic manifold]] forms a Poisson algebra. On a symplectic manifold, every real-valued  function ''H'' on the manifold induces a vector field ''X<sub>H</sub>'', the [[Hamiltonian vector field]].  Then, given any two smooth functions ''F'' and ''G'' over the symplectic manifold, the Poisson bracket may be defined as:

:<math>\{F,G\}=dG(X_F) = X_F(G)\,</math>.

This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as

:<math>X_{\{F,G\}}=[X_F,X_G]\,</math>

where [,] is the [[Lie derivative]].  When the symplectic manifold is '''R'''<sup>2''n''</sup> with the standard symplectic structure, then the Poisson bracket takes on the well-known form

:<math>\{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}.</math>

Similar considerations apply for [[Poisson manifold]]s, which generalize symplectic manifolds by allowing the symplectic bivector to be rank deficient.