極座標系での速度·加速度ベクトル

2次元極座標系

デカルト座標系と極座標系の対応を
$${\bm{r}(t)=\left(\begin{array}{}r(t)\cos\theta(t)\\r(t)\sin\theta(t)\end{array}\right)^{\text{T}}}$$とする。
以下引数を省略し、$${\dfrac{\mathrm{d}\bigcirc}{\mathrm{d}t}}$$を$${\dot{\bigcirc}}$$などと書く。
極座標系の基底を$${\bm{e}_{r},\,\bm{e}_{\theta}}$$とすると、そのデカルト座標成分表示は
$${\bm{e}_{r}=\left(\begin{array}{}\cos\theta\\\sin\theta\end{array}\right)^{\text{T}},\,\bm{e}_{\theta}=\left(\begin{array}{}-\sin\theta\\\cos\theta\end{array}\right)^{\text{T}}}$$
である。

極座標系の基底の時間微分はそれぞれ

$$
\begin{array}{}
\dot{\bm{e}}_{r}&=&
\left(\begin{array}{}
-\dot{\theta}\sin\theta\\
\dot{\theta}\cos\theta
\end{array}\right)^{\text{T}}\\
&=&\dot{\theta}\bm{e}_{\theta}\\\\
\dot{\bm{e}}_{\theta}&=&
\left(\begin{array}{}
-\dot{\theta}\cos\theta\\
-\dot{\theta}\sin\theta
\end{array}\right)^{\text{T}}\\
&=&-\dot{\theta}\bm{e}_{r}
\end{array}
$$

となる。したがって速度$${\bm{v}}$$と加速度$${\bm{a}}$$は

$$
\begin{array}{}
\bm{v}&=&\dot{\bm{r}}\\
&=&\dot{r}\bm{e}_{r}+r\dot{\bm{e}}_{r}\\
&=&\dot{r}\bm{e}_{r}
+r\dot{\theta}\bm{e}_{\theta}\\\\
\bm{a}&=&\dot{\bm{v}}=\ddot{\bm{r}}\\
&=&\ddot{r}\bm{e}_{r}
+\dot{r}\dot{\bm{e}}_{r}
+\dot{r}\dot{\theta}\bm{e}_{\theta}
+r\ddot{\theta}\bm{e}_{\theta}
+r\dot{\theta}\dot{\bm{e}}_{\theta}\\
&=&\ddot{r}\bm{e}_{r}
+2\dot{r}\dot{\theta}\bm{e}_{\theta}
+r\ddot{\theta}\bm{e}_{\theta}
-r\dot{\theta}^{2}\bm{e}_{r}\\
&=&\left(\ddot{r}-r\dot{\theta}^{2}\right)
\bm{e}_{r}
+\left(r\ddot{\theta}+2\dot{r}\dot{\theta}
\right)\bm{e}_{\theta}
\end{array}
$$

となる。

3次元極座標系

デカルト座標系と極座標系の対応を
$${\bm{r}(t)=\left(\begin{array}{}r(t)\sin\theta(t)\cos\varphi(t)\\r(t)\sin\theta(t)\sin\varphi(t)\\r(t)\cos\theta(t)\end{array}\right)^{\text{T}}}$$
とする。
極座標系の基底を$${\bm{e}_{r},\,\bm{e}_{\theta},\,\bm{e}_{\varphi}}$$とすると、そのデカルト座標成分表示は

$$
\begin{array}{}
\bm{e}_{r}&=&\left(\begin{array}{}
\sin\theta\cos\varphi\\
\sin\theta\sin\varphi\\
\cos\theta\end{array}\right)^{\text{T}}\\
\bm{e}_{\theta}&=&\left(\begin{array}{}
\cos\theta\cos\varphi\\
\cos\theta\sin\varphi\\
-\sin\theta\end{array}\right)^{\text{T}}\\
\bm{e}_{\varphi}&=&\left(\begin{array}{}
-\sin\varphi\\\cos\varphi\\0
\end{array}\right)^{\text{T}}
\end{array}
$$

である。極座標系の基底の時間微分はそれぞれ

$$
\begin{array}{}
\dot{\bm{e}}_{r}&=&
\left(\begin{array}{}
\dot{\theta}\cos\theta\cos\varphi\\
\dot{\theta}\cos\theta\sin\varphi\\
-\dot{\theta}\sin\theta
\end{array}\right)^{\text{T}}
+\left(\begin{array}{}
-\dot{\varphi}\sin\theta\sin\varphi\\
\dot{\varphi}\sin\theta\cos\varphi\\
0\end{array}\right)^{\text{T}}\\
&=&\dot{\theta}\bm{e}_{\theta}
+\dot{\varphi}\sin\theta\bm{e}_{\varphi}\\\\
\dot{\bm{e}}_{\theta}&=&
\left(\begin{array}{}
-\dot{\theta}\sin\theta\cos\varphi\\
-\dot{\theta}\sin\theta\sin\varphi\\
-\dot{\theta}\cos\theta
\end{array}\right)^{\text{T}}
+\left(\begin{array}{}
-\dot{\varphi}\cos\theta\sin\varphi\\
\dot{\varphi}\cos\theta\cos\varphi\\
0\end{array}\right)^{\text{T}}\\
&=&-\dot{\theta}\bm{e}_{r}
+\dot{\varphi}\cos\theta\bm{e}_{\varphi}\\\\
\dot{\bm{e}}_{\varphi}&=&
\left(\begin{array}{}
-\dot{\varphi}\cos\varphi\\
-\dot{\varphi}\sin\varphi\\0
\end{array}\right)^{\text{T}}\\
&=&-\dot{\varphi}\sin\theta\bm{e}_{r}
-\dot{\varphi}\cos\theta\bm{e}_{\theta}
\end{array}
$$

となる。したがって速度$${\bm{v}}$$と加速度$${\bm{a}}$$は

$$
\begin{array}{}
\bm{v}&=&\dot{\bm{r}}\\
&=&\dot{r}\bm{e}_{r}+r\dot{\bm{e}}_{r}\\
&=&\dot{r}\bm{e}_{r}
+r\dot{\theta}\bm{e}_{\theta}
+r\dot{\varphi}\sin\theta\bm{e}_{\varphi}
\\\\
\bm{a}&=&\dot{\bm{v}}\\
&=&\ddot{r}\bm{e}_{r}
+\dot{r}\dot{\bm{e}}_{r}
+\dot{r}\dot{\theta}\bm{e}_{\theta}
+r\ddot{\theta}\bm{e}_{\theta}
+r\dot{\theta}\dot{\bm{e}}_{\theta}
+\dot{r}\dot{\varphi}\sin\theta
\bm{e}_{\varphi}\\&&
+r\ddot{\varphi}\sin\theta\bm{e}_{\varphi}
+r\dot{\theta}\dot{\varphi}\cos\theta
\bm{e}_{\varphi}
+r\dot{\varphi}\sin\theta
\dot{\bm{e}}_{\varphi}\\
&=&\ddot{r}\bm{e}_{r}
+\dot{r}(\dot{\theta}\bm{e}_{\theta}
+\dot{\varphi}\sin\theta\bm{e}_{\varphi})
+\dot{r}\dot{\theta}\bm{e}_{\theta}
+r\ddot{\theta}\bm{e}_{\theta}\\&&
+r\dot{\theta}(-\dot{\theta}\bm{e}_{r}
+\dot{\varphi}\cos\theta\bm{e}_{\varphi})
+\dot{r}\dot{\varphi}\sin\theta
\bm{e}_{\varphi}
+r\ddot{\varphi}\sin\theta\bm{e}_{\varphi}
\\&&
+r\dot{\theta}\dot{\varphi}\cos\theta
\bm{e}_{\varphi}
+r\dot{\varphi}\sin\theta
(-\dot{\varphi}\sin\theta\bm{e}_{r}
-\dot{\varphi}\cos\theta
\bm{e}_{\theta})\\
&=&\left(\ddot{r}-r\dot{\theta}^{2}
-r\dot{\varphi}^{2}\sin^{2}\theta\right)
\bm{e}_{r}\\&&
+\left(2\dot{r}\dot{\theta}+r\ddot{\theta}
-r\dot{\varphi}^{2}\sin\theta\cos\theta
\right)\bm{e}_{\theta}\\&&
+\left(2\dot{r}\dot{\varphi}\sin\theta
+r\ddot{\varphi}\sin\theta
+2r\dot{\theta}\dot{\varphi}\cos\theta
\right)\bm{e}_{\varphi}
\end{array}
$$