A proof of Bertrand's theorem using the theory of isochronous potentials [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
David Rojas
Universitat de Girona

Given a field of forces in the Euclidean space which is central and attractive there always exist circular periodic solutions. In 1873 Bertrand proved the following result: among all central fields of forces in the Euclidean space there are only two exceptional cases (the harmonic oscillator and the Newtonian potential) in which all solutions close to the circular motions are also periodic. Besides the original proof, nowadays there are several methods of proofs.

In this talk we present a connection between Bertrand's theorem and the theory of planar isochronous potential centers. In particular, we are interested in the families of potentials of the type $V_{\lambda}(x)=\frac{1}{2}x^2+\lambda\Phi(x),\,x>0,$ where $\Phi\in C^2(0,+\infty)$ is a given function. We prove that there are only two isochronous families of this type and we derive a new proof of Bertrand's theorem from it. The proof is more or less direct in the analytic scenario but there are some subtleties when the central field of forces is only $C^1$.