極座標系
en: polar coordinates system
ここでは 特に2次元のを あつかう
aka. 円座標系 (en: circular coodinates system)
formula
位置ベクトル: $ \bm{r}=r\hat{\bm{r}}
where $ r = \|\bm{r}\|
これは当然
角速度: $ \omega(t)=\frac{\mathrm{d}}{\mathrm{d}t}\theta(t) = \dot{\theta}(t)
where $ \theta = \|\bm{\theta}\|
基底（ベクトル）の時間微分
$ \frac{\mathrm{d}}{\mathrm{d}t}\hat{\bm{r}} = \frac{\mathrm{d}\theta}{\mathrm{d}t}\hat{\bm{\theta}} = \omega\hat{\bm{\theta}}
$ \frac{\mathrm{d}}{\mathrm{d}t}\hat{\bm{\theta}} = -\frac{\mathrm{d}\theta}{\mathrm{d}t}\hat{\bm{r}} = -\omega\hat{\bm{r}}
+θ方向にτ/4回転: R(τ/4)
速度: $ \bm{v} = \frac{\mathrm{d}}{\mathrm{d}t}(r\hat{\bm{r}}) = \frac{\mathrm{d}r}{\mathrm{d}t}\hat{\bm{r}} + r\frac{\mathrm{d}\hat{\bm{r}}}{\mathrm{d}t} = \begin{pmatrix} \frac{\mathrm{d}}{\mathrm{d}t}r(t) \\ r(t) \frac{\mathrm{d}\theta(t)}{\mathrm{d}t} \end{pmatrix} = \begin{pmatrix} \frac{\mathrm{d}}{\mathrm{d}t}r(t) \\ r(t) \omega(t) \end{pmatrix}
加速度: $ \bm{a} = \frac{\mathrm{d}^2}{{\mathrm{d}t}^2} \begin{pmatrix} r(t) \\ \theta(t) \end{pmatrix} = \begin{pmatrix} \frac{\mathrm{d}^2r(t)}{{\mathrm{d}t}^2} - r(t)\left(\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}\right)^2 \\ \frac{1}{r(t)}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left(r(t)\right)^2\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}\right) \end{pmatrix} = \begin{pmatrix} \frac{\mathrm{d}^2r(t)}{{\mathrm{d}t}^2} - r(t)\left(\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}\right)^2 \\ 2\frac{\mathrm{d}r(t)}{\mathrm{d}t}\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}+r(t)\frac{\mathrm{d}^2\theta(t)}{{\mathrm{d}t}^2} \end{pmatrix}
$ = \begin{pmatrix} \frac{\mathrm{d}^2r(t)}{{\mathrm{d}t}^2} - r(t)\left(\omega(t)\right)^2 \\ \frac{1}{r(t)}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left(r(t)\right)^2\omega(t)\right) \end{pmatrix} = \begin{pmatrix} \frac{\mathrm{d}^2r(t)}{{\mathrm{d}t}^2} - r(t)\left(\omega(t)\right)^2 \\ 2\frac{\mathrm{d}r(t)}{\mathrm{d}t}\omega(t)+r(t)\frac{\mathrm{d}\omega(t)}{{\mathrm{d}t}} \end{pmatrix}
$ \begin{pmatrix} F_r \\ F_\theta \end{pmatrix} = R(-\theta)\begin{pmatrix} F_x \\ F_y \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} F_x \\ F_y \end{pmatrix}
$ \begin{pmatrix} x \\ y \end{pmatrix} = R(\theta)\begin{pmatrix} r \\ \theta \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} r \\ \theta \end{pmatrix}
$ \dfrac{\bm{F}}{m} = \ddot{\bm{r}} = \begin{pmatrix} \ddot{r}(t) - r(t)({\dot{\theta}(t)})^2 \\ \frac{1}{r(t)}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left(r(t)\right)^2\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}\right) \end{pmatrix}
等速円運動
条件: $ \dot{r}=0, \ddot{r}=0, \ddot{\theta}=0
結果: $ \bm{v}=r\omega\hat\bm{\theta}, \bm{a}=-r\omega^2\hat\bm{r}
ベクトル解析
∇: $ \nabla = \frac{\partial}{\partial{r}}\hat{r}+\frac{1}{r}\frac{\partial}{\partial{\theta}}\hat{\theta} = \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{1}{r}\frac{\partial}{\partial \theta} \end{pmatrix}
nabla: grad、勾配ベクトル場
$ \nabla = \begin{pmatrix} \dfrac{\partial}{\partial x} \\ \dfrac{\partial}{\partial y} \end{pmatrix} = \begin{pmatrix} \cos\theta & - \dfrac{\sin\theta}{r} \\ \sin\theta & \dfrac{\cos\theta}{r} \end{pmatrix} \begin{pmatrix} \dfrac{\partial}{\partial r} \\ \dfrac{\partial}{\partial \theta} \end{pmatrix}
$ = R(\theta) \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{r} \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{\partial}{\partial \theta} \end{pmatrix} = R(\theta) \operatorname{diag}\left(1,\frac{1}{r}\right) \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{\partial}{\partial \theta} \end{pmatrix}
ref.
回転
grad