泊松括号
给定函数 $u(q, p, t)$ 和 $v(q, p, t)$，定义泊松括号为
\begin{equation}
\left\{u, v\right\} = \sum_i \frac{\partial u}{\partial q_i} \frac{\partial v}{\partial p_i} - \frac{\partial v}{\partial q_i} \frac{\partial u}{\partial p_i}
\end{equation}
容易证明泊松括号满足
\begin{equation}
\left\{v, u\right\} = - \left\{u, v\right\}
\end{equation}
泊松括号与守恒量
对任意不显含时的物理量 $\omega (q,p)$ 都有
\begin{equation}
\dot \omega = \sum_i \left( \frac{\partial \omega}{\partial q_i} \dot q_i + \frac{\partial \omega}{\partial p_i} \dot p_i \right)
= \sum_i \left( \frac{\partial \omega}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial H}{\partial q_i} \frac{\partial \omega}{\partial p_i} \right)
= \left\{\omega, H\right\}
\end{equation}
所以若泊松括号恒等于零，则该物理量守恒．
同理，当 $\omega (q,p,t)$ 显含时间时有
\begin{equation}
\dot \omega = \left\{\omega, H\right\} + \frac{\partial \omega}{\partial t}
\end{equation}
量子力学中的对易算符对应泊松括号．该式对应量子力学中的算符平均值演化方程．