Completely stuck on a homework problem
The orbital motion of a satellite in earth orbit is given by the following equations:
$\ddot{r}(t)-\dot{\theta}(t)^2 r(t) = \frac{\mu}{r(t)^2}+a_r(t), \mu=\omega_0 ^2 R^3$ (i)
$a_i(t)=r(t)\ddot{\theta}(t)+2\dot{r}(t)\dot{\theta}(t)$ (ii)
where $a_i(t)$ and $a_r(t)$ are the control accelerations, $r(t)$ and $\theta(t)$ are the radial and transverse components of the position vector and $\mu$ is a constant given for a circular orbit and $\omega_0$ is the angular velocity and $R$ is the radius of the circular orbit and therefore a constant.
What I'm trying to do:
A)
Assuming that the outputs of interest are $r(t)$ and $\theta(t)$, obtain a state-space representation for equations (i)-(ii).
B)
Show that $\tilde{r}(t)=R,\tilde{\dot{r}}(t)=0,\tilde{\theta}(t)=\omega_0t+\theta_0,\tilde{\dot{\theta}}(t)=\omega_0$ is the state trajectory associated to null accelerations $\tilde{a}_i(t)=\tilde{a}_r(t)=0$ when the initial condition is $r(0)=R,\dot{r}(0)=0,\theta(0)=\theta_0,\dot{\theta}(0)=\omega_0$ (this state trajectory is the circuar orbit)
C)
Classify the system. If the system is linear, describe it in a compact matrix representation using matrices $A$, $B$, $C$, $D$. Otherwise, linearize it around the state trajectory corresponding to the circular orbit.
What I've tried to do:
Laplace transform, but doesn't work due to $L\{\dot{\theta}^2\}=???$ I don't know how to even search for a methodology or how to categorize the problem in terms of differentiation techniques.
Tried to find some algebraic trick to make the problem simpler to be able to perform Laplace transform or some differentiation technique I know, but can't seem to find any.
Any kind of direction/recommended reading to get me moving in the right direction is greatly appreciated!
Thanks for reading.