Consider two-body central force problem in polar co-ordinates $r,θ$.
Corresponding 2nd order differential equation is obtained by using conservation of angular momentum. This equation is :
$ d^2r/dt^2=l^2/(m^2r^3)−GM/r^2 $
$r(t)$ is the radial position of particle (of mass $m$) as a function of time $t$. $l$ is angular momentum which is constant. $G$ is gravitational constant and $M$ is mass of the heavier body, assumed to be at rest at the origin of co-ordinate system i.e. at $(r,θ)=(0,0)$
I want to solve above non-linear differential equation; it is non-linear since dependent variable $r$ has powers -3 and -2 on RHS.
Can I use 4-order Runge-Kutta method to solve this equation ?
Extra Note: Actually we have two different 2-order differential equations (coupled) : one for $r$ and another for $θ$. Conservation of angular momentum de-couples them and reduces to one equation given above. Also if we try to solve the above 1-Dim equation analytically, we end up with a solution of the form $t(r)$ i.e. time is function of $r$. So we have to invert that into $r(t)$. This inversion process can be extremely difficult in practice. Please see standard textbook on classical mechanics e.g. by Goldstein (Chapter 3).
Here Initial conditions should be on $r$ and $dr/dt$, if we want to solve numerically. But if we want to solve analytically we need initial conditions as Total Energy and Angular momentum of the mass $m$. I am confused here: do I need use all initial conditions i.e. energy, angular momentum, $r(t=0)$ and $dr/dt$ at $t=0$ ?