Determination of the apsidal angles and Bertrand’s theorem

We derive an expression for the determination of the apsidal angles that holds good for arbitrary central potentials. This formula can be useful in the calculation of precession rates. Then we discuss under what conditions the apsidal angles remain independent of the mechanical energy and angular momentum in the central force problem and, as a consequence, an alternative and nonperturbative proof of Bertrand’s theorem is obtained.

Figure 1

General form of the effective potential energy.

Figure 2

Contour for the integration of Eq. (45).