What are the Components of drag force in spherical polar coordinates? [closed]

Question

How to decompose the drag force in three components of spherical polar coordinates?

Since
$ F =\frac{1}{2}\rho C A v^2 $

or in vector form :

$\vec F = \vert F\vert \hat F = \frac{1}{2}\rho C A v^2 \hat v = \frac{1}{2}\rho C A v^2 \frac {\hat v}{\vert {\vec v}\vert} = \frac{1}{2}\rho C A v \hat v$

$\dot{r}= \vec v$ = $r\hat{r}$ + $r\dot{\phi}\hat{\theta}$ + $r\dot{\phi}\sin{\theta}\hat{\phi}, $ in spherical polar coordinates
$v^2$ = $r^2 + r^2{\dot{\theta}^2}+ r^2\dot{\phi}\sin^2{\theta}$

can one write $ F_\theta =\frac{1}{2}\rho C A (r^2{\dot{\theta}^2}) ? $ ; $ F_\phi =\frac{1}{2}\rho C A (r^2\dot{\phi}\sin^2{\theta}) ? $

For a particle are moving along the azimuthal direction through the plasma is the force $ F = F_\phi =\frac{1}{2}\rho C A (r^2\dot{\phi}\sin^2{\theta}) ? $

Answer 1

Q1: The formula given for $F$ is not a vector equation, so I don't think I can get vector components out of it.

Q2: Note that $v^2=\mathbf{v}\cdot\mathbf{v}$, and use $\hat{\mathbf{r}}\cdot \hat{\mathbf{r}} = \hat{\boldsymbol{\theta}}\cdot \hat{\boldsymbol{\theta}} = \hat{\boldsymbol{\phi}}\cdot \hat{\boldsymbol{\phi}} = 1$ in the conventional spherical coordinate system. This means that $$v^2=\dot r^2+r^2\dot\theta^2+r^2\dot\phi ^2\sin^2\theta.$$

Q3: Motion in the azimuthal direction fixes $r=R$ and $\theta = \theta_0$, so we have $$v^2=R^2\dot\phi ^2\sin^2\theta_0, $$ which makes perfect geometrical sense. Just think of a particle constrained to move at constant latitude on a sphere.

Note: $\phi$ here denotes the azimuthal angle and $\theta$ the polar angle. The line element is $$d\mathbf{s} = dr\:\hat{\mathbf{r}} + r\:d\theta\:\hat{\boldsymbol{\theta}} + r\sin \theta \:d\phi \:\hat{\boldsymbol{\phi}} $$