# Bertrand theorem - central forces

Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ either $F(r)=b/r^2$.

Here are my questions : is this statement (or a related one) a mathematical theorem ? If yes, is there a nice reference ?

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This MO question (on 2- and 3-body problems under various force laws) seems related: mathoverflow.net/questions/35980/… . Perhaps the references there to Arnolʹd's work may be relevant. – Joseph O'Rourke Jan 7 '11 at 21:57
The proof is contained in Arnold's book Mathematical methods of classical mechanics in the form of the sequence of exercises, part 1, chapter 2, paragraph 8. The idea is to look at circular trajectories and then to say that all trajectories close to them are also close (sorry for my English;)). This is classical perturbation argument. Though, reading a response by Will Heierman, I now see that we have to be accurate. – Olga Jun 9 '15 at 9:32

Bertrand's theorem is essentially correct as stated, but most of the proofs are a little weak at one point - the assumption of stability of "nearly circular orbits" - that the apsidal angles are continuously dependent and have a finite limit as the perturbations approach 0. This is the argument that Bertrand used to assert that, since they were always rationally commensurate with one revolution, the angle had to be constant.
In general this may not be true, because perturbations may be spirals instead of rosettes (the apsidal angle integral may diverge). However, if one imposes the "Lagrange stability condition" (that r^3 times the force law increases with r), then the orbits must be rosettes, continuous dependence can be established, and the power rules can be "calculated". The existence of non-circular bounded orbits can then be used to show the restriction of the Lagrange stability condition can be removed from the statement of the result.
A final remark: this can all be shown rigorously following Bertrand's brilliant argument using mathematical techniques known at the time (Riemann integration theory) his original paper was published(1853).

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there is an extensive mathematical literature on Bertrand's theorem and generalizations; some pointers:

Hamiltonian Systems Admitting a Runge-Lenz Vector and an Optimal Extension of Bertrand's Theorem to Curved Manifolds, A. Ballesteros et al., COMMUNICATIONS IN MATHEMATICAL PHYSICS, 290, 1033 (2009).

Multifold Kepler systes: Dynamical systems all of whose bounded trajectories are closed, T. Iwai and N. Katayama, JOURNAL OF MATHEMATICAL PHYSICS, 36, 1790 (1995); 35, 2914 (1994).

Extension of Bertrand's theorem and factorization of the radial Schrodinger equation, Z.B. Wu and J.Y. Zeng, JOURNAL OF MATHEMATICAL PHYSICS, 39, 5253 (1998).

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Thank you for the answer. I will try to find what I am looking for in those papers. On the other hand, if someone has a a ready reference for a simple mathematical statement and its proof (even with stronger assumptions than needed), I would really appreciate it ! – camomille Jan 8 '11 at 9:30

Since you say central forces in the title, this looks to me like the standard Bertrand theorem described in Goldstein's Classical Mechanics (Section 3.6, proof in Appendix A). Or is there something nonstandard about your assumptions that I have missed?

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There is no nonstandard assumptions in my question. Actually, most references I found seemed to be physical references. I have no easy access to the above book but if it contains a mathematical proof I will try to find it. – camomille Jan 8 '11 at 19:12
Here's another reference that can be read on Google books: Boccaletti & Pucacco, Theory of orbits. – Hans Lundmark Jan 9 '11 at 15:06