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 ?

Thank you for any answer.

Mathematical methods of classical mechanicsin 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