# 帕松括號

## 正則坐標

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

## 運動方程

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

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

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

## 運動常數

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

## 定義

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）：${\displaystyle \omega }$，這就是說${\displaystyle d\omega =0}$且當其視一個映射${\displaystyle \omega :\xi \in \mathrm {vect} [M]\rightarrow i_{\xi }\omega \in \Lambda ^{1}[M]}$${\displaystyle \omega }$有逆映射${\displaystyle {\tilde {\omega }}:\Lambda ^{1}[M]\rightarrow \mathrm {vect} [M]}$。 這裏${\displaystyle d}$是流形M上內蘊的外導數運算，而${\displaystyle i_{\xi }\theta }$內乘縮並運算，在1-形式${\displaystyle \theta }$這等價於${\displaystyle \theta (\xi )}$

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

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

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

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

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

## 李代數

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

${\displaystyle X_{q}=\sum _{i}X^{i}(q){\frac {\partial }{\partial q^{i}}}}$

${\displaystyle P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}}$

${\displaystyle \{P_{X},P_{Y}\}(q,p)=\sum _{i}\sum _{j}\{X^{i}(q)\;p_{i},Y^{j}(q)\;p_{j}\}}$
${\displaystyle =\sum _{ij}p_{i}Y^{j}(q){\frac {\partial X^{i}}{\partial q^{j}}}-p_{j}X^{i}(q){\frac {\partial Y^{j}}{\partial q^{i}}}}$
${\displaystyle =-\sum _{i}p_{i}\;[X,Y]^{i}(q)}$
${\displaystyle =-P_{[X,Y]}(q,p)\,}$