球坐标系速度和加速度的推导

做法之一

球坐标基矢按直角坐标展开：

$$\begin{cases} \hat{r} &= \sin\theta\cos\varphi \hat{x} +\sin\theta\sin\varphi \hat{y} +\cos\theta \hat{z}\\\\ \hat{\theta} &= \cos\theta\cos\varphi \hat{x} +\cos\theta\sin\varphi \hat{y} -\sin\theta \hat{z}\\\\ \hat{\varphi} &= -\sin\varphi \hat{x} +\cos \varphi \hat{y} \end{cases}$$

球坐标系的基矢不像直角坐标的基矢一样是固定的，而是随坐标的改变而改变：

$$\begin{cases} \hat{r} &= \hat{r}(\theta,\varphi) \\\\ \hat{\theta} &= \hat{r}(\theta,\varphi) \\\\ \hat{\varphi} &= \hat{r}(\varphi) \end{cases}$$

所球坐标系中的坐标的表达式为：

$$\vec{r} = r \hat{r}$$

速度的表达式为：

$$\dot{\vec{r}} = \dot{r}\hat{r} + r \dot{\hat{r}}$$

而：

$$\begin{split} \dot{\hat{r}} =& \dot{\theta}(\cos\theta\cos\varphi \hat{x} +\cos\theta\sin\varphi \hat{y} -\sin\theta \hat{z}) \\\\ &+\dot{\varphi}(-\sin\theta\sin\varphi \hat{x} +\sin\theta\cos \varphi \hat{y}) \\\\ =& \dot{\theta}\hat{\theta} + \dot{\varphi}\sin\theta \hat{\varphi} \end{split}$$

所以：

$$\dot{\vec{r}} = \dot{r}\hat{r} + r \dot{\hat{r}} = \dot{r}\hat{r} +r\dot{\theta}\hat{\theta} + r\dot{\varphi}\sin\theta \hat{\varphi}$$

加速度的表达式为：

$$\ddot{\vec{r}} = \ddot{r}\hat{r}+\dot{r}\dot{\hat{r}}+\dot{r}\dot{\hat{r}}+ r \ddot{\hat{r}} = \ddot{r}\hat{r}+2\dot{r}\dot{\hat{r}}++ r \ddot{\hat{r}}$$

而：

$$\ddot{\hat{r}} = \ddot{\theta}\hat{\theta}+\dot{\theta}\dot{\hat{\theta}} + \ddot{\varphi}\sin\theta\hat{\varphi} +\dot{\theta}\dot{\varphi}\cos\theta\hat{\varphi} + \dot{\varphi}\sin\theta \dot{\hat{\varphi}}$$

其中 $\dot{\hat{\theta}}$ 和 $\dot{\hat{\varphi}}$ 为：

$$\begin{split} \dot{\hat{\theta}} =& \dot{\theta}(-\sin\theta\cos\varphi \hat{x} -\sin\theta\sin\varphi \hat{y} -\cos\theta \hat{z}) \\\\ &+\dot{\varphi}(-\cos\theta\sin\varphi \hat{x} +\cos\theta\cos \varphi \hat{y}) \\\\ =& -\dot{\theta}\hat{r} + \dot{\varphi}\cos\theta\hat{\varphi} \end{split}$$ $$\begin{split} \dot{\hat{\varphi}} =&\dot{\varphi}(-\cos\varphi \hat{x}-\sin\varphi\hat{y}) = \dot\varphi (-\sin\theta\hat{r}-\cos\theta\hat{\theta}) \end{split}$$

则：

$$\begin{split} \ddot{\hat{r}} =& -\dot{\theta}^2\hat{r}+\dot{\theta}\dot{\varphi}\cos\theta\hat{\varphi}+ \ddot{\theta}\hat{\theta}+ \ddot{\varphi}\sin\theta\hat{\varphi} +\dot{\theta}\dot{\varphi}\cos\theta\hat{\varphi} -\dot{\varphi}^2\sin^2\theta\hat{r}-\dot{\varphi}^2\sin\theta\cos\theta\hat{\theta} \\\\ =& (-\dot{\theta}^2-\dot{\varphi}^2\sin^2\theta)\hat{r} +(\ddot{\theta}-\dot{\varphi}^2\sin\theta\cos\theta)\hat\theta +(2\dot{\theta}\dot{\varphi}\cos\theta+\ddot{\varphi}\sin\theta)\hat{\varphi} \end{split}$$

最终得加速度的表达式为：

$$\begin{split} \ddot{\vec{r}} = & \ddot{r}\hat{r}+2\dot{r}\dot{\theta}\hat{\theta}+2\dot{r}\dot{\varphi}\sin\theta\hat{\varphi}+ r \ddot{\hat{r}} \\\\ =&(\ddot{r}-r\dot{\theta}^2-r\dot{\varphi}^2\sin^2\theta)\hat{r} \\\\ &+(2\dot{r}\dot{\theta}+r\ddot{\theta}-r\dot{\varphi}^2\sin\theta\cos\theta)\hat{\theta} \\\\ &+(2\dot{r}\dot{\varphi}\sin\theta+2r\dot{\theta}\dot{\varphi}\cos\theta+r\ddot{\varphi}\sin\theta)\hat{\varphi} \end{split}$$

做法之二

坐标基矢之间的变换：

$$\left(\begin{matrix} \hat{e}_r\\ \hat{e}_{\theta}\\ \hat{e}_{\phi}\\ \end{matrix}\right) = \left(\begin{matrix} \sin\theta\cos\phi &\sin\theta\sin\phi &\cos\theta \\ \cos\theta\cos\phi &\cos\theta\sin\phi &-\sin\theta \\ -\sin\phi &\cos\phi & 0 \end{matrix}\right) \left(\begin{matrix} \hat{e}_x\\ \hat{e}_y\\ \hat{e}_z\\ \end{matrix}\right)$$

坐标基矢之间的逆变换：

$$\left(\begin{matrix} \hat{e}_x\\ \hat{e}_y\\ \hat{e}_z\\ \end{matrix}\right) = \left(\begin{matrix} \sin\theta\cos\phi &\cos\theta\cos\phi &-\sin\phi \\ \sin\theta\sin\phi &\cos\theta\sin\phi &\cos\phi \\ \cos\theta &-\sin\theta & 0 \end{matrix}\right) \left(\begin{matrix} \hat{e}_r\\ \hat{e}_{\theta}\\ \hat{e}_{\phi}\\ \end{matrix}\right)$$

坐标之间的变换：

$$\begin{cases} r = \sqrt{x^2+y^2+z^2}\\ \theta = \arccos\frac{z}{ \sqrt{x^2+y^2+z^2}} \\ \phi = \arctan\frac{y}{x} \end{cases}$$

坐标之间的逆变换：

$$\begin{cases} x &= r\sin\theta\cos\phi \\ y &=r\sin\theta\sin\phi\\ z &= r\cos\theta \end{cases}$$

质点位置的表达式：

$$\vec{r} = x \hat{e}_x +y\hat{e}_y + z\hat{e}_z$$

将坐标的变换和基矢的变换代入可得：

$$\vec{r} = r \hat{e}_r$$

速度的表达式为：

$$\vec{v} = \dot{x} \hat{e}_x + \dot{y} \hat{e}_y + \dot{z} \hat{e}_z$$

其中

$$\begin{cases} \dot{x} &= \dot{r}\sin\theta\cos\phi +r\dot{\theta}\cos\theta\cos\phi-r\dot{\phi}\sin\theta\sin\phi \\ \dot{y} &=\dot{r}\sin\theta\sin\phi+r\dot{\theta}\cos\theta\sin\phi+r\dot{\phi}\sin\theta\cos\phi \\ \dot{z} &= \dot{r}\cos\theta-r\dot{\theta}\sin\theta \end{cases}$$

代入变换关系可得：

$$\vec{v} = \dot{r}\hat{e}_r + r\dot{\theta}\hat{e}_{\theta} + r\dot{\phi}\sin\theta \hat{e}_{\phi}$$

加速度表达式为：

$$\vec{a} = \ddot{x} \hat{e}_x + \ddot{y} \hat{e}_y + \ddot{z} \hat{e}_z$$

代入变换关系可得

$$\vec{a}=(\ddot{r}-r\dot{\theta}^2-r\dot{\phi}^2\sin^2\theta)\hat{e}_{r} \\\\ +(2\dot{r}\dot{\theta}+r\ddot{\theta}-r\dot{\phi}^2\sin\theta\cos\theta)\hat{e}_{\theta} \\\\ +(2\dot{r}\dot{\phi}\sin\theta+2r\dot{\theta}\dot{\phi}\cos\theta+r\ddot{\phi}\sin\theta)\hat{e}_{\varphi}$$