# 正則變換生成函數

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

## 生成函數列表

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

## 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

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

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

${\displaystyle \mathbf {p} ={\frac {\partial G_{2}}{\partial \mathbf {q} }}}$(7)
${\displaystyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}}$(8)
${\displaystyle {\mathcal {K}}={\mathcal {H}}+{\frac {\partial G_{2}}{\partial t}}}$(9)

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

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

${\displaystyle \mathbf {P} =\mathbf {P} (\mathbf {q} ,\ \mathbf {p} ,\ t)}$(10)

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

${\displaystyle \mathbf {Q} =\mathbf {Q} (\mathbf {q} ,\ \mathbf {p} ,\ t)}$(11)

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

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

## 第三型生成函數

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

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

## 第四型生成函數

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

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

## 實例 1

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

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

## 實例 2

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

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

## 實例 3

${\displaystyle {\mathcal {H}}=aP^{2}+bQ^{2}}$

${\displaystyle {\mathcal {H}}={\frac {1}{2q^{2}}}+{\frac {p^{2}q^{4}}{2}}}$(12)

${\displaystyle P=pq^{2}}$(13)
${\displaystyle Q=-{\frac {1}{q}}}$(14)

${\displaystyle {\mathcal {H}}={\frac {Q^{2}}{2}}+{\frac {P^{2}}{2}}}$

${\displaystyle {\frac {\partial G_{3}}{\partial Q}}=-P}$

${\displaystyle {\frac {\partial G_{3}}{\partial Q}}=-{\frac {p}{Q^{2}}}}$

${\displaystyle G_{3}(p,\ Q)={\frac {p}{Q}}}$

${\displaystyle q=-{\frac {\partial G_{3}}{\partial p}}={\frac {-1}{Q}}}$