Definition:Poisson Bracket

Definition
Let $A = \map A{x,\family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ and $B = \map B {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ be real functions, dependent on canonical variables.

Then:


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

is called the Poisson Bracket of functions $A$ and $B$.