泊松括號

正則坐標

$\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right] .$

运动方程

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac{\partial f}{\partial t} + \frac {\partial f}{\partial p} \frac {\mathrm{d}p}{\mathrm{d}t} + \frac {\partial f}{\partial q} \frac {\mathrm{d}q}{\mathrm{d}t}.$

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac{\partial f}{\partial t} + \frac {\partial f}{\partial q} \frac {\partial H}{\partial p} - \frac {\partial f}{\partial p} \frac {\partial H}{\partial q} = \frac{\partial f}{\partial t} +\{f,H\}.$

$\frac{\mathrm{d}}{\mathrm{d}t} f= \left(\frac{\partial }{\partial t} - \{\,H, \cdot\,\}\right)f.$

运动常数

$0 = \frac {\mathrm{d}}{\mathrm{d}t} f(p,q) = \frac {\partial f}{\partial p} \frac {\mathrm{d}p}{\mathrm{d}t} + \frac {\partial f}{\partial q} \frac {\mathrm{d}q}{\mathrm{d}t} = \frac {\partial f}{\partial q} \frac {\partial H}{\partial p} - \frac {\partial f}{\partial p} \frac {\partial H}{\partial q} = \{f,H\}$

定义

M 是一個辛流形，即流形上帶有一個辛形式的非退化2-形式）：$\omega$，这就是说 $d\omega = 0$ 且当其视一个映射 $\omega: \xi \in \mathrm{vect}[M] \rightarrow i_\xi \omega \in \Lambda^1[M]$$\omega$ 有逆映射 $\tilde{\omega}: \Lambda^1[M] \rightarrow \mathrm{vect}[M]$。 这里 $d$ 是流形 M 上内蕴的外导数运算，而 $i_\xi \theta$内乘缩并运算，在 1-形式$\theta$ 这等价于 $\theta(\xi)$

$i_{[v, w]} \omega = d(i_v i_w \omega) + i_v d(i_w \omega) - i_w d(i_v \omega) - i_w i_v d\omega ,\,$

$\{f,g\} = i_{\tilde{\omega}(df)} dg = - i_{\tilde{\omega}(dg)} df = -\{g,f\}\,$

$\{f,\{g,h\}\} + \{g,\{h,f\}\} + \{h,\{f,g\}\} = 0\,$

$\{f,gh\} = \{f,g\}h + g\{f,h\}\,$

$\{f,\{g,h\}\} - \{g,\{f,h\}\} = \{\{f,g\},h\}\,$

李代數

$\{P_X,P_Y\}=-P_{[X,Y]} \,$

$X_q=\sum_i X^i(q) \frac{\partial}{\partial q^i}$

$P_X(q,p)=\sum_i X^i(q) \;p_i$

$\{P_X,P_Y\}(q,p)= \sum_i \sum_j \{X^i(q) \;p_i, Y^j(q)\;p_j \}$
$=\sum_{ij} p_i Y^j(q) \frac {\partial X^i}{\partial q^j} - p_j X^i(q) \frac {\partial Y^j}{\partial q^i}$
$= - \sum_i p_i \; [X,Y]^i(q)$
$= - P_{[X,Y]}(q,p) \,$