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My professor and I have been working on a proof of Bertrand's Theorem using perturbative methods. We have arrived at a solution yielding $1/r^3$, which we had presumed to be an incorrect result. While I'm new to his research, I have been obsessing over finding reconciliation or a SPoF.

However, after reading the last comment on the first reply to this particular SE post, I am reconsidering this result: An intuitive proof of Bertrand's theorem. Can somebody elaborate on what @mmesser314 is talking about? I haven't seen a perturbation-based derivation lead to a $1/r^3$ result in the literature I've encountered. I'd really appreciate it.

Qmechanic
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Jeesubmunu
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  • It is not clear what you are saying. If your result is that a potential $1/r^3$ yields closed orbit that is clearly wrong. Another way of looking at Bertrand's theorem is that for the two solutions (harmonic oscillator and $1/r$) there are additional conservation laws. For gravity this conservation is related to the Runge-Lenz vector. – lcv Jan 23 '20 at 09:13
  • Coming to @mmesser314 's comment you're talking about I think that's a typo, but he/she may be able to say more. – lcv Jan 23 '20 at 09:17

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I have been studying Bertrand's theorem recently. I won't comment on @mmesser314 answer but a classic reference for a proof using perturbation theory is [1]. A more recent paper working with $r(\theta)$ instead of trying to solve for $\theta(t)$, and also looking at oscillations about circular orbits is [2].

[1] Brown, Lowell S. (1978). “Forces giving no orbit precession”. In: American Journal of Physics 46.9, pp. 930– 931. doi: 10.1119/1.11519.

[2] Chin, Siu A. (2015). “A truly elementary proof of Bertrand’s theorem”. In: American Journal of Physics 83, p. 320.