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.