Definition:Poisson Bracket

Definition
Let $A = A \left({x, \langle y_i \rangle_{1 \mathop \le i \mathop \le n}, \langle p_i \rangle_{1 \mathop \le i \mathop \le n} } \right)$ and $B = B\left({ x, \langle y_i \rangle_{ 1 \mathop \le i \mathop \le n}, \langle p_i \rangle_{ 1 \mathop \le i \mathop \le n} } \right)$ be real functions, dependent on canonical variables.

Then:


 * $\displaystyle \left[{ A, B } \right] = \sum_{i \mathop = 1}^n \left({ \frac{ \partial A }{ \partial y_i } \frac{ \partial B }{ \partial p_i } - \frac{ \partial B }{ \partial y_i } \frac{ \partial A }{ \partial p_i }  } \right)$

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