Definition:Poisson Bracket

Definition
For $n \in \N$, let:


 * $\map A {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$


 * $\map B {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$

be real functions, dependent on canonical variables.

The Poisson bracket of functions $A$ and $B$ is defined as:


 * $\ds \sqbrk {A, B} := \sum_{i \mathop = 1}^n \paren {\frac {\partial A} {\partial y_i} \frac {\partial B} {\partial p_i} - \frac {\partial B} {\partial y_i} \frac {\partial A} {\partial p_i} }$

where the notation $\dfrac {\partial A} {\partial y_i}$ denotes partial differentiation.