The Bertrand theorem revisited

The Bertrand theorem, which states that the only power-law central potentials for which the bounded trajectories are closed are 1/r² and r², is analyzed using the Poincaré-Lindstedt perturbation method. This perturbation method does not generate secular terms and correctly incorporates the effect of nonlinearities on the nature of periodic solutions. The requirement that the orbits be closed implies that the theorem holds in each order of the expansion.