# 作用量-角度坐标

（重定向自作用量-角度座標

## 導引

### 保守的哈密頓量系統

${\displaystyle {\mathcal {H}}(\mathbf {q} ;\ \mathbf {p} )=a_{\mathcal {H}}}$

${\displaystyle \mathbf {p} ={\frac {\partial W}{\partial \mathbf {q} }}}$
${\displaystyle \mathbf {Q} ={\frac {\partial W}{\partial \mathbf {P} }}}$

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

${\displaystyle {\dot {\mathbf {P} }}=-{\frac {\partial {\mathcal {K}}}{\partial Q}}=0,\!}$

${\displaystyle \mathbf {P} =\mathbf {a} }$

${\displaystyle W(\mathbf {q} ;\ \mathbf {a} )=\sum _{i=1}^{n}\ W_{i}(q_{i};\ \mathbf {a} )}$

${\displaystyle p_{i}={\frac {\partial W_{i}(q_{i};\ \mathbf {a} )}{\partial q_{i}}}}$
${\displaystyle Q_{i}=\sum _{j=1}^{n}\ {\frac {\partial W_{j}(q_{j};\ \mathbf {a} )}{\partial a_{i}}}}$

### 週期性運動

${\displaystyle J_{i}\equiv \oint p_{i}dq_{i}}$

${\displaystyle W(\mathbf {q} ;\ \mathbf {J} )=\sum _{i=1}^{n}\ W_{i}(q_{i};\ \mathbf {J} )}$

${\displaystyle w_{i}\equiv {\frac {\partial W}{\partial J_{i}}}=\sum _{j=1}^{n}\ {\frac {\partial W_{j}(q_{j};\ \mathbf {J} )}{\partial J_{i}}}}$

${\displaystyle W(\mathbf {w} ;\ \mathbf {J} )=\sum _{i=1}^{n}\ W_{i}(w_{i};\ \mathbf {J} )}$

${\displaystyle {\mathcal {K}}'(\mathbf {w} ;\ \mathbf {J} )={\mathcal {H}}(\mathbf {q} ;\ \mathbf {p} )=a_{\mathcal {H}}}$

${\displaystyle -{\dot {J}}_{i}={\frac {\partial {\mathcal {K}}'}{\partial w_{i}}}=0}$

${\displaystyle \nu _{i}(\mathbf {J} )={\dot {w}}_{i}={\frac {\partial {\mathcal {K}}'}{\partial J_{i}}}}$

${\displaystyle w_{i}=\nu _{i}t+\beta _{i}}$

### 運動頻率

${\displaystyle T_{i}=\oint dt=\oint {\frac {dq_{i}}{\dot {q_{i}}}}=\oint {\cfrac {dq_{i}}{\ \ {\cfrac {\partial {\mathcal {H}}}{\partial p_{i}}}\ \ }}}$

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}=\sum _{j=1}^{n}{\frac {\partial {\mathcal {K}}'}{\partial J_{j}}}{\frac {\partial J_{j}}{\partial p_{i}}}=\sum _{j=1}^{n}\nu _{j}{\frac {\partial J_{j}}{\partial p_{i}}}}$

${\displaystyle J_{j}\equiv \oint p_{j}dq_{j}=p_{j}\oint dq_{j}=p_{j}\ell }$

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}=\sum _{j=1}^{n}\nu _{j}\delta _{ij}\,\ell =\nu _{i}\,\ell }$

${\displaystyle T_{i}=\oint {\frac {dq_{i}}{\nu _{i}(\mathbf {J} )\,\ell }}={\frac {1}{\nu _{i}}}}$

${\displaystyle w_{i}=w_{i}(\mathbf {q} ;\ \mathbf {J} )}$

${\displaystyle \delta w_{i}=\sum _{j=1}^{n}{\frac {\partial w_{i}}{\partial q_{j}}}dq_{j}}$

${\displaystyle T=m_{1}T_{1}+m_{2}T_{2}}$

${\displaystyle T=\sum _{i=1}^{n}m_{i}T_{i}}$

${\displaystyle \Delta w_{i}=\nu _{i}m_{i}T_{i}=\oint \sum _{j=1}^{n}{\frac {\partial w_{i}}{\partial q_{j}}}dq_{j}=\oint \sum _{j=1}^{n}\sum _{k=1}^{n}{\frac {\partial ^{2}W_{k}(q_{k};\ \mathbf {J} )}{\partial q_{j}\ \partial J_{i}}}dq_{j}}$

${\displaystyle \Delta w_{i}={\frac {d}{dJ_{i}}}\oint \sum _{j=1}^{n}\sum _{k=1}^{n}{\frac {\partial W_{k}(q_{k};\ \mathbf {J} )}{\partial q_{j}}}dq_{j}={\frac {d}{dJ_{i}}}\oint \sum _{j=1}^{n}p_{j}dq_{j}={\frac {d}{dJ_{i}}}\sum _{j=1}^{n}m_{j}J_{j}=m_{i}}$

${\displaystyle \nu _{i}(\mathbf {J} )={\frac {1}{T}}}$

### 傅立葉級數

${\displaystyle q_{k}=\sum _{s_{1}=-\infty }^{\infty }\sum _{s_{2}=-\infty }^{\infty }\ldots \sum _{s_{N}=-\infty }^{\infty }A_{s_{1},s_{2},\ldots ,s_{N}}^{k}e^{i2\pi s_{1}w_{1}}e^{i2\pi s_{2}w_{2}}\ldots e^{i2\pi s_{N}w_{N}}}$

${\displaystyle q_{k}=\sum _{s_{k}=-\infty }^{\infty }e^{i2\pi s_{i}w_{i}}}$

## 基本規則總結

1. 計算作用量變數 ${\displaystyle J_{i}}$
2. 用作用量變數表示原本哈密頓量。
3. 取哈密頓量關於作用量變數的導數。這樣，可以求得頻率 ${\displaystyle \nu _{i}}$

## 參考文獻

• H. Goldstein, (1980) Classical Mechanics, 2nd. Ed., Addison-Wesley. ISBN 0-201-02918-9. pg. 457-477.