# 作用量-角度坐标

## 導引

### 保守的哈密頓量系統

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

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

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

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

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

$W(\mathbf{q};\ \mathbf{a})=\sum_{i=1}^n\ W_i(q_i;\ \mathbf{a})$

$p_i=\frac{\partial W_i(q_i;\ \mathbf{a})}{\partial q_i}$
$Q_{i}=\sum_{j=1}^n\ \frac{\partial W_j(q_j;\ \mathbf{a})}{\partial a_{i}}$

### 週期性運動

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

$W(\mathbf{q};\ \mathbf{J})=\sum_{i=1}^n\ W_i(q_i;\ \mathbf{J})$

$w_{i} \equiv \frac{\partial W}{\partial J_i}=\sum_{j=1}^n\ \frac{\partial W_j(q_j;\ \mathbf{J})}{\partial J_{i}}$

$W(\mathbf{w};\ \mathbf{J})=\sum_{i=1}^n\ W_i(w_i;\ \mathbf{J})$

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

$- \dot{J}_i=\frac{\partial \mathcal{K}'}{\partial w_i}=0$

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

$w_{i} = \nu_{i} t + \beta_{i}$

### 運動頻率

$T_{i}=\oint dt=\oint \frac{dq_i}{\dot{q_i}}=\oint \cfrac{dq_i}{\ \ \cfrac{\partial \mathcal{H}}{\partial p_i}\ \ }$

$\frac{\partial \mathcal{H}}{\partial p_i}=\sum_{j=1}^n \frac{\partial \mathcal{K}'}{\partial J_j}\frac{\partial J_j}{\partial p_i}=\sum_{j=1}^n \nu_j \frac{\partial J_j}{\partial p_i}$

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

$\frac{\partial \mathcal{H}}{\partial p_i}=\sum_{j=1}^n \nu_j \delta_{ij}\, \ell=\nu_i\,\ell$

$T_{i}=\oint \frac{dq_i}{\nu_i(\mathbf{J})\,\ell}=\frac{1}{\nu_i}$

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

$\delta w_{i}=\sum_{j=1}^n \frac{\partial w_i}{\partial q_j} dq_j$

$T=m_1T_1+m_2T_2$

$T=\sum_{i=1}^n m_iT_i$

$\Delta w_{i} = \nu_{i}m_i T_i=\oint \sum_{j=1}^n \frac{\partial w_{i}}{\partial q_{j}} dq_{j} =\oint\sum_{j=1}^n \sum_{k=1}^n \frac{\partial^2 W_k(q_k;\ \mathbf{J})}{\partial q_{j}\ \partial J_{i}}dq_{j}$

$\Delta w_{i}=\frac{d}{dJ_{i}} \oint \sum_{j=1}^n \sum_{k=1}^n \frac{\partial W_k(q_k;\ \mathbf{J})}{\partial q_{j}} dq_{j} = \frac{d}{dJ_{i}} \oint \sum_{j=1}^n p_{j} dq_{j} = \frac{d}{dJ_{i}}\sum_{j=1}^n m_jJ_j=m_i$

$\nu_{i}(\mathbf{J}) = \frac{1}{T}$

### 傅立葉級數

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

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

## 基本規則總結

1. 計算作用量變數 $J_{i}$
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 $\nu_{i}$

## 參考文獻

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