Talk:Bertrand's theorem

Non-perturbative demonstration
--Guerinsylvie (talk) 10:45, 5 August 2010 (UTC): Good morning, i am new on english-WP and french-locutor ; Arnold, mathematical methods in classical mechanics, ed Mir. It uses a "transmutation" of the force of Newton ( in Principia). See also, Arnold : on Barrow, Huygens, Newton, Hooke.
 * it exists another demonstration in :
 * and another :
 * More, if you want, best-respects.

In the first paragraph, it states, "an inverse-square central force" and the first equation is just an inverse force. Shouldn't the equation be changed to


 * $$ V(\mathbf{r}) = \frac{-k}{r^2},$$?

LDCorey (talk) 04:50, 29 September 2014 (UTC)

No LDCorey, V is here for the potential, potential is (a curve) integral of the force. Or, the force is a gradient of the potential. To inverse square *force*, the potential proportional to 1/r correctly corresponds. — Preceding unsigned comment added by 89.176.67.241 (talk) 23:33, 12 July 2016 (UTC)

Closed orbits vs Stable orbits
It is well known that power potentials with n > 3 do have a stable, circular (therefore closed) orbit, thus invalidating the "necessary condition" written in the derivation that "Here we show that stable, exactly closed orbits can be produced only with an inverse-square force or radial harmonic oscillator potential." In my opinion, what is lacking is that all orbits are like so for those potentials, whereas the circular orbit is the outlier for the other powers. I am making that correction in the article but would appreciate feedback anyways. Bcpicao (talk) 18:26, 16 July 2022 (UTC)