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 R=Mm1r1+m2r2
where M=m1+m2. Now, using r=r1−r2, r1=R+Mm2rr2=R−Mm1r
Note that the momentum P=MR˙=0 implies that the centre of mass is stationary (R=0)
Looking at Kinetic Energy, T=21(m1r˙12+m2r˙22)=21[m1(R˙+Mm2r˙)2+m1(R˙−Mm1r˙)2]=21MR˙2+21μr˙2
where μ=Mm1m2, 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, T=21μr˙2.
Equation of Motion
Let our central force and it's potential be F=−f(r)r^U=U(r)
The angular momentum of the system (w.r.t. CM) will become, L=(m1r12+m2r22)ω=μr2θ˙
Using r˙=r˙r^+rθ˙θ^, T=21μ(r˙2+r2θ˙2)
Thus, the total energy, EE=T+U=21μ(r˙2+r2θ˙2)+U(r)=21μr˙2+2μr2L2+U(r)=21μr˙2+Ueff
where Ueff=2μr2L2+U(r)
Note: I've written rotational kinetic energy in terms of L is because θ term is messy and L is constant.
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Great Work
Okay, this is dead. I have no idea what I intended to say after this. Good day!