## Position, Velocity and Acceleration in Cartesian and Spherical Coordinates

The position vector of a point in Cartesian coordinates is
$\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$

Expressed in spherical polar coordinates
$(r, \theta \ \phi)$

$x= r sin \theta cos \phi$

$y= r sin \theta sin \phi$

$z= r cos \theta$

Hence
$\mathbf{r} = r sin \theta cos \phi \mathbf{i} + r sin \theta sin \phi \mathbf{j} + r cos \theta \mathbf{k}$

\begin{aligned} \mathbf{v} &= \frac{d \mathbf{r}}{dt} \\ &= (\dot{r} sin \theta cos \phi + r \dot{\theta} cos \theta cos \phi - r \dot{\phi} sin \theta sin \phi ) \mathbf{i} \\ &+ (\dot{r} sin \theta sin \phi + r \dot{\theta} cos \theta cos \phi - r \dot{\phi} sin \theta \cos \phi ) \mathbf{j} \\ &+ (\dot{r} cos \theta - r \dot{\theta} sin \theta ) \mathbf{k} \\ &= \dot{r} (sin \theta cos \phi \mathbf{i} + sin \theta sin \phi \mathbf{j} + cos \phi \mathbf{k}) \\ &+ r \dot{\theta} ( cos \theta cos \phi \mathbf{i} + cos \theta cos \phi \mathbf{j} - sin \theta \mathbf{k}) \\ &+ r \dot{\phi} sin \theta (sin \phi \mathbf{i} - \cos \phi \mathbf{j}) \\ &= \dot{r} \mathbf{e_r} + r \dot{\theta} \mathbf{e_\theta} + r \dot{\phi} sin \theta \mathbf{e_\phi} \end{aligned}

\begin{aligned} \mathbf{a} &= \frac{d \mathbf{v}}{dt} \\ &= \ddot{r} \mathbf{e_r} + \dot{r} \dot{\mathbf{e_r}} \\ &+ \dot{r} \dot{\theta} \mathbf{e_\theta} + r \ddot{\theta} \mathbf{e_\theta} + r \dot{\theta} \dot{\mathbf{e_\theta}} \\ &+ \dot{r}\dot{\phi} sin \theta \mathbf{e_\phi} + r \ddot{\phi} sin \theta \mathbf{e_\phi} +r \dot{\phi} \dot{\theta} cos \theta \mathbf{e_\phi} +r \dot{\phi} sin \theta \dot{\mathbf{e_\phi}} \\ &= \ddot{r} \mathbf{e_r} + \dot{r} (\dot{\theta} {\mathbf{e_\theta}} + \dot{\phi} sin \theta \mathbf{e_{\phi}}) \\ &+ \dot{r} \dot{\theta} \mathbf{e_\theta} + r \ddot{\theta} \mathbf{e_\theta} + r \dot{\theta} (- \dot{\theta} \mathbf{e_r} + \dot{\phi} cos \theta \mathbf{e_{\phi}}) \\ &+ \dot{r}\dot{\phi} sin \theta \mathbf{e_\phi} + r \ddot{\phi} sin \theta \mathbf{e_\phi} +r \dot{\phi} \dot{\theta} cos \theta \mathbf{e_\phi} +r \dot{\phi} sin \theta (-\dot{\phi} sin \theta \mathbf{e_r} - \dot{\phi} cos \theta \mathbf{e_\theta}) \\ &= \ddot{r} \mathbf{e_r} + \dot{r} \dot{\theta} {\mathbf{e_\theta}} +\dot{r} \dot{\phi} sin \theta \mathbf{e_{\phi}} \\ &+ \dot{r} \dot{\theta} \mathbf{e_\theta} + r \ddot{\theta} \mathbf{e_\theta} + - r \dot{\theta} \dot{\theta} \mathbf{e_r} + r \dot{\theta} \dot{\phi} cos \theta \mathbf{e_{\phi}} \\ &+ \dot{r}\dot{\phi} sin \theta \mathbf{e_\phi} + r \ddot{\phi} sin \theta \mathbf{e_\phi} +r \dot{\phi} \dot{\theta} cos \theta \mathbf{e_\phi} +r \dot{\phi} sin \theta - r \dot{\phi} sin \theta \dot{\phi} sin \theta \mathbf{e_r} - r \dot{\phi} sin \theta \dot{\phi} cos \theta \mathbf{e_\theta} \\ &= (\ddot{r} - r \dot{\theta}^2 -r \dot{\phi}^2 sin \theta) \mathbf{e_r} \\ &+ (2 \dot{r} \dot{\theta}+ r \ddot{\theta} -r \dot{\phi}^2 sin \theta cos \theta )\mathbf{e_\theta} \\ &+ (2r \dot{\phi} sin \theta +2r \dot{\theta} \dot{\phi} cos \theta+ r \ddot{\phi} sin \theta ) \mathbf{e_\phi} \end{aligned}