Poisson bracket

Definition

The Poisson bracket is a bilinear operator on an associative algebra $A$ $\{-,-\}:A\times A\to A$ that satisfies the following conditions:

• Antisymmetric: $\{x,y\}=-\{y,x\}$ for all $x,y\in A$.
• Jacobi Indentity: $\{x,\{y,z\}\}+\{y,\{z,x\}\}+\{z,\{x,y\}\}=0$
• Liebniz Rule: $\{x,yz\}=\{x,y\}z+\{x,z\}y$

We can think of the Poisson bracket as a Lie bracket with an extra condition.