# 正則變換

## 定義

${\displaystyle q_{i}=q_{i}(Q_{1},\ Q_{2},\ \dots ,\ Q_{N},\ t)\ ,\qquad \qquad \qquad \qquad i=1,\ 2,\ 3,\ \dots ,\ N}$

${\displaystyle q_{i}=q_{i}(Q_{1},\ Q_{2},\ \dots ,\ Q_{N},\ P_{1},\ P_{2},\ \dots ,\ P_{N},\ t)\ ,\qquad \qquad \qquad \qquad i=1,\ 2,\ 3,\ \dots ,\ N}$
${\displaystyle p_{i}=p_{i}(Q_{1},\ Q_{2},\ \dots ,\ Q_{N},\ P_{1},\ P_{2},\ \dots ,\ P_{N},\ t)\ ,\qquad \qquad \qquad \qquad i=1,\ 2,\ 3,\ \dots ,\ N}$

${\displaystyle {\dot {\mathbf {q} }}=~~{\frac {\partial {\mathcal {H}}}{\partial \mathbf {p} }}}$
${\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {q} }}}$

${\displaystyle {\dot {\mathbf {Q} }}=~~{\frac {\partial {\mathcal {K}}}{\partial \mathbf {P} }}}$
${\displaystyle {\dot {\mathbf {P} }}=-{\frac {\partial {\mathcal {K}}}{\partial \mathbf {Q} }}}$

## 實際用處

${\displaystyle {\mathcal {H}}={\mathcal {H}}(\mathbf {q} ,\ \mathbf {p} ,\ t)}$

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q_{i}}}=0}$

${\displaystyle {\dot {p}}_{i}=-{\frac {\partial {\mathcal {H}}}{\partial q_{i}}}=0}$

## 生成函數方法

${\displaystyle \delta \int _{t_{1}}^{t_{2}}\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {H}}(\mathbf {q} ,\ \mathbf {p} ,\ t)\right]dt=0}$
${\displaystyle \delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-{\mathcal {K}}(\mathbf {Q} ,\ \mathbf {P} ,\ t)\right]dt=0}$

${\displaystyle \sigma \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {H}}(\mathbf {q} ,\ \mathbf {p} ,\ t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-{\mathcal {K}}(\mathbf {Q} ,\ \mathbf {P} ,\ t)+{\frac {dG}{dt}}}$

${\displaystyle \sigma \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {H}}\right]=\mathbf {P} '\cdot {\dot {\mathbf {Q} }}'-{\mathcal {K}}\,'+{\frac {dG\,'}{dt}}}$

${\displaystyle {\frac {\partial {\mathcal {K}}}{\partial \mathbf {P} }}=\alpha {\frac {\partial {\mathcal {K}}\,'}{\partial \mathbf {P} '}}=\alpha {\dot {\mathbf {Q} }}'={\dot {\mathbf {Q} }}}$
${\displaystyle {\frac {\partial {\mathcal {K}}}{\partial \mathbf {Q} }}=\beta {\frac {\partial {\mathcal {K}}\,'}{\partial \mathbf {Q} '}}=-\beta {\dot {\mathbf {P} }}'=-{\dot {\mathbf {P} }}}$
${\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {H}}=\alpha \beta (\mathbf {P} '\cdot {\dot {\mathbf {Q} }}'-{\mathcal {K}}\,'+{\frac {dG\,'}{dt}})=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-{\mathcal {K}}+{\frac {dG}{dt}}}$ （1）

### 第一型生成函數

${\displaystyle G=G_{1}(\mathbf {q} ,\ \mathbf {Q} ,\ t)}$

${\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {H}}(\mathbf {q} ,\ \mathbf {p} ,\ t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-{\mathcal {K}}(\mathbf {Q} ,\ \mathbf {P} ,t)+{\frac {\partial G_{1}}{\partial t}}+{\frac {\partial G_{1}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}}$

${\displaystyle \mathbf {p} =~~{\frac {\partial G_{1}}{\partial \mathbf {q} }}}$ （2）
${\displaystyle \mathbf {P} =-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}}$ （3）
${\displaystyle {\mathcal {K}}={\mathcal {H}}+{\frac {\partial G_{1}}{\partial t}}}$ （4）

${\displaystyle 2N+1}$ 個方程式設定了變換${\displaystyle (\mathbf {q} ,\ \mathbf {p} )\rightarrow (\mathbf {Q} ,\ \mathbf {P} )}$ ，步驟如下：

${\displaystyle \mathbf {p} =\mathbf {p} (\mathbf {q} ,\ \mathbf {Q} ,\ t)}$

${\displaystyle \mathbf {Q} =\mathbf {Q} (\mathbf {q} ,\ \mathbf {p} ,\ t)}$ （5）

${\displaystyle \mathbf {P} =\mathbf {P} (\mathbf {q} ,\ \mathbf {Q} ,\ t)}$

${\displaystyle \mathbf {P} =\mathbf {P} (\mathbf {q} ,\ \mathbf {p} ,\ t)}$ （6）

${\displaystyle 2N}$ 個函數方程式（5）、（6），可以逆算出${\displaystyle 2N}$ 個函數方程式

${\displaystyle \mathbf {q} =\mathbf {q} (\mathbf {Q} ,\ \mathbf {P} ,\ t)}$
${\displaystyle \mathbf {p} =\mathbf {p} (\mathbf {Q} ,\ \mathbf {P} ,\ t)}$

${\displaystyle {\mathcal {K}}={\mathcal {K}}(\mathbf {Q} ,\ \mathbf {P} ,\ t)}$

### 第二型生成函數

${\displaystyle G=-\mathbf {Q} \cdot \mathbf {P} +G_{2}(\mathbf {q} ,\ \mathbf {P} ,\ t)}$

${\displaystyle \mathbf {p} ={\frac {\partial G_{2}}{\partial \mathbf {q} }}}$ ，
${\displaystyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}}$
${\displaystyle {\mathcal {K}}={\mathcal {H}}+{\frac {\partial G_{2}}{\partial t}}}$

### 第三型生成函數

${\displaystyle G=\mathbf {q} \cdot \mathbf {p} +G_{3}(\mathbf {p} ,\mathbf {Q} ,t)}$

${\displaystyle \mathbf {q} =-{\frac {\partial G_{3}}{\partial \mathbf {p} }}}$
${\displaystyle \mathbf {P} =-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}}$
${\displaystyle {\mathcal {K}}={\mathcal {H}}+{\frac {\partial G_{3}}{\partial t}}}$

### 第四型生成函數

${\displaystyle G=\mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} +G_{4}(\mathbf {p} ,\mathbf {P} ,t)}$

${\displaystyle \mathbf {q} =-{\frac {\partial G_{4}}{\partial \mathbf {p} }}}$
${\displaystyle \mathbf {Q} =~~{\frac {\partial G_{4}}{\partial \mathbf {P} }}}$
${\displaystyle {\mathcal {K}}={\mathcal {H}}+{\frac {\partial G_{4}}{\partial t}}}$

### 實例1

${\displaystyle G_{1}=\mathbf {q} \cdot \mathbf {Q} }$

${\displaystyle \mathbf {p} =~~{\frac {\partial G_{1}}{\partial \mathbf {q} }}=\mathbf {Q} }$
${\displaystyle \mathbf {P} =-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}=-\mathbf {q} }$

${\displaystyle {\mathcal {K}}(\mathbf {Q} ,\ \mathbf {P} ,\ t)={\mathcal {H}}(\mathbf {q} ,\ \mathbf {p} ,\ t)}$

### 實例2

${\displaystyle G_{2}\equiv \mathbf {g} (\mathbf {q} ;\ t)\cdot \mathbf {P} }$

${\displaystyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}=\mathbf {g} (\mathbf {q} ;\ t)}$

## 不變量

### 辛條件

${\displaystyle {\boldsymbol {\xi }}^{T}=[q_{1},\ q_{2},\ q_{3},\ \dots ,\ q_{N},\ p_{1},\ p_{2},\ p_{3},\ \dots ,\ p_{N}]}$

${\displaystyle {\dot {\boldsymbol {\xi }}}={\boldsymbol {\Omega }}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {\xi }}}}}$

${\displaystyle {\dot {\boldsymbol {\Xi }}}={\boldsymbol {\Omega }}{\frac {\partial {\mathcal {K}}}{\partial {\boldsymbol {\Xi }}}}}$

${\displaystyle {\boldsymbol {\Xi }}={\boldsymbol {\Xi }}({\boldsymbol {\xi }},\ t)}$ 關於時間${\displaystyle t}$ 的導數，

${\displaystyle {\dot {\boldsymbol {\Xi }}}=\mathbf {M} {\dot {\boldsymbol {\xi }}}+{\frac {\partial {\boldsymbol {\Xi }}}{\partial t}}}$

${\displaystyle \mathbf {M} {\dot {\boldsymbol {\xi }}}+{\frac {\partial {\boldsymbol {\Xi }}}{\partial t}}={\boldsymbol {\Omega }}{\frac {\partial {\mathcal {K}}}{\partial {\boldsymbol {\Xi }}}}={\boldsymbol {\Omega }}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {\Xi }}}}+{\boldsymbol {\Omega }}{\frac {\partial ^{2}G_{1}}{\partial {\boldsymbol {\Xi }}\ \partial t}}}$  ;

${\displaystyle \mathbf {M} {\dot {\boldsymbol {\xi }}}={\boldsymbol {\Omega }}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {\Xi }}}}}$

${\displaystyle {\mathcal {H}}={\mathcal {H}}({\boldsymbol {\xi }})}$ ，所以，

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {\Xi }}}}={\frac {\partial {\boldsymbol {\xi }}}{\partial {\boldsymbol {\Xi }}}}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {\xi }}}}=(\mathbf {M} ^{-1})^{T}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {\xi }}}}=-(\mathbf {M} ^{-1})^{T}{\boldsymbol {\Omega }}{\dot {\boldsymbol {\xi }}}}$

${\displaystyle \mathbf {M} =-{\boldsymbol {\Omega }}(\mathbf {M} ^{-1})^{T}{\boldsymbol {\Omega }}}$

${\displaystyle \mathbf {M} ^{T}=-{\boldsymbol {\Omega }}\mathbf {M} ^{-1}{\boldsymbol {\Omega }}}$
${\displaystyle \mathbf {M} ^{T}{\boldsymbol {\Omega }}={\boldsymbol {\Omega }}\mathbf {M} ^{-1}}$

${\displaystyle \mathbf {M} ^{T}{\boldsymbol {\Omega }}\mathbf {M} ={\boldsymbol {\Omega }}}$

### 基本帕松括號不變量

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

${\displaystyle {\big [}f,\ g{\big ]}_{\boldsymbol {\xi }}=\left({\frac {\partial f}{\partial {\boldsymbol {\xi }}}}\right)^{T}{\boldsymbol {\Omega }}\ {\frac {\partial g}{\partial {\boldsymbol {\xi }}}}}$

${\displaystyle {\big [}q_{i},\ q_{j}{\big ]}_{\boldsymbol {\xi }}={\big [}p_{i},\ p_{j}{\big ]}_{\boldsymbol {\xi }}=0}$
${\displaystyle {\big [}q_{i},\ p_{j}{\big ]}_{\boldsymbol {\xi }}=-{\big [}p_{i},\ q_{j}{\big ]}_{\boldsymbol {\xi }}=\delta _{ij}}$

${\displaystyle {\big [}{\boldsymbol {\xi }},\ {\boldsymbol {\xi }}{\big ]}_{\boldsymbol {\xi }}={\boldsymbol {\Omega }}}$

${\displaystyle {\big [}{\boldsymbol {\Xi }},\ {\boldsymbol {\Xi }}{\big ]}_{\boldsymbol {\xi }}=\left({\frac {\partial {\boldsymbol {\Xi }}}{\partial {\boldsymbol {\xi }}}}\right)^{T}{\boldsymbol {\Omega }}\ {\frac {\partial {\boldsymbol {\Xi }}}{\partial {\boldsymbol {\xi }}}}}$

${\displaystyle M={\frac {\partial {\boldsymbol {\Xi }}}{\partial {\boldsymbol {\xi }}}}}$

${\displaystyle {\big [}{\boldsymbol {\Xi }},\ {\boldsymbol {\Xi }}{\big ]}_{\boldsymbol {\xi }}=\mathbf {M} ^{T}{\boldsymbol {\Omega }}\mathbf {M} }$

${\displaystyle {\big [}{\boldsymbol {\Xi }},\ {\boldsymbol {\Xi }}{\big ]}_{\boldsymbol {\xi }}={\boldsymbol {\Omega }}}$

### 帕松括號不變量

{\displaystyle {\begin{aligned}{\big [}f,\ g{\big ]}_{\boldsymbol {\xi }}&=\left({\frac {\partial f}{\partial {\boldsymbol {\xi }}}}\right)^{T}{\boldsymbol {\Omega }}\ {\frac {\partial g}{\partial {\boldsymbol {\xi }}}}\\&=\left({\frac {\partial {\boldsymbol {\Xi }}}{\partial {\boldsymbol {\xi }}}}{\frac {\partial f}{\partial {\boldsymbol {\Xi }}}}\right)^{T}{\boldsymbol {\Omega }}\ {\frac {\partial {\boldsymbol {\Xi }}}{\partial {\boldsymbol {\xi }}}}{\frac {\partial g}{\partial {\boldsymbol {\Xi }}}}\\&=\left({\frac {\partial f}{\partial {\boldsymbol {\Xi }}}}\right)^{T}M^{T}{\boldsymbol {\Omega }}M{\frac {\partial g}{\partial {\boldsymbol {\Xi }}}}\ \ _{\circ }\\\end{aligned}}}

${\displaystyle {\big [}f,\ g{\big ]}_{\boldsymbol {\xi }}=\left({\frac {\partial f}{\partial {\boldsymbol {\Xi }}}}\right)^{T}{\boldsymbol {\Omega }}\ {\frac {\partial g}{\partial {\boldsymbol {\Xi }}}}={\big [}f,\ g{\big ]}_{\boldsymbol {\Xi }}}$

## 參考文獻

1. ^ Goldstein, Herbert. Classical Mechanics 3rd. United States of America: Addison Wesley. 1980: pp. 384. ISBN 0201657023 （英语）.