I saw a question a few days ago which referred to Bertrand's theorem. So, I now know that stable, closed orbits only occur when the potential function is $\frac{-k}{r}$ or $\tfrac{1}{2}kr^2$.
- If we relax the conditions to stable but not necessarily closed orbits then what are the possibilities?
- If we restrict consideration to power law forces $kr^n$, for what $n$ are stable orbits possible?
My memory was that for $n \geq -2$ there are stable orbits but for $n \leq -3$ there are not. I forget the cases between $-2$ and $-3$.
I believe that a circular orbit is possible for all $n$ but consider a slight perturbation to that orbit. For the well known $n = -2$ case, we will get a ellipse with low eccentricity. If my memory is right then for $n = -1$, the orbit will be stable just not closed. For $n = -3$, the orbit will not be stable and spiral in or out after the perturbation.
Am I right on the power law forces?
I have tried searching here and elsewhere but failed to find my answer.
Clarification: note that I am not interested only in circular orbits. The primary question is whether they are stable if perturbed but further information on the nature of stable orbits for central forces would be very welcome.