Equations of Motion of a Satellite I'm working on a control problem where I need to know the equations of motion for a satellite orbiting the earth about a central axis. Using Newton's second law, I was able to show that 
$\ddot{r}=\dfrac{-Gm_e}{r^2}-r\dot{\theta^2}$,
but I have not been able to get the equations for $\ddot{\theta}$. I know it should be 
$\ddot{\theta}=-2\dfrac{\dot{r}\dot{\theta}}{r}$,
but my physics skills are rusty. 
An explanation for where this comes from would be greatly appreciated. 
 A: $\ddot{r}=\dfrac{-Gm_e}{r^2}-r\dot{\theta^2}$,
$\ddot{\theta}=-2\dfrac{\dot{r}\dot{\theta}}{r}$,
It is now classical. To derive satellite orbits we use two equations of equilibrium in curvilinear coordinates. To get both components, twice differentiate the vector $ p=r e^{ i \theta} $ to get the radial and circumferential force components with respect to Sun-Earth line as position vector.
$ \dfrac {\dot{p}} {e^{ i \theta}} = \dot{r}  + i r {\dot{\theta}}  $ 
$ \dfrac {\ddot{p}} {e^{ i \theta}} =  (\ddot{r} -r\dot{\theta^2}) + i ( r \ddot{\theta} +2 \dot{r} \dot{\theta})  $
Radial equilibrium is where Newton famously introduced the inverse law of force as you gave and,
Circumferential equilibrium is :
$$ r \ddot{\theta}+ 2 \dot{r}\dot{\theta}= 0$$
as no force acts in this direction. This  on integration gives conserved constant of integration $h$ , the angular momentum 
$$ r^2 \dot{\theta} $$
tallying with Kepler's constant swept area rate Law. Introducing $1/r$ as a variable, elliptic orbits with sun focus are obtained.
