Definition:Poisson Bracket

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

Then


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

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