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

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}
$$