I would like to share some insight on how different trajectories can form in a central field. I'll use laws of Newtonian gravitation and I would limit myself to examples in gravitation.
Two Body Central Force
In a two body system, we normally assume the heavier to be fixed. I will not. Also, I will proceed by taking the motion of the reduced mass of the system.
Going down to basic equations, coordinate of C.M. becomes
where . Now, using ,
Note that the momentum implies that the centre of mass is stationary ()
Looking at Kinetic Energy,
where , the reduced mass of the system. It's time for us to shift to the centre of mass frame of reference.
The Centre of Mass reference frame
From now on, all standard variables will be referred on the C.M. frame.
In C.M. frame, for example, our kinetic energy becomes, .
Equation of Motion
Let our central force and it's potential be
The angular momentum of the system (w.r.t. CM) will become,
Thus, the total energy,
Note: I've written rotational kinetic energy in terms of is because term is messy and is constant.
This will be updated