The Moving Singularities of the Perturbation Expansion of the Classical Kepler Problem

Abstract

The convergence properties of the perturbation expansion for the periodic solution of the classical Kepler problem are studied as a function of the perturbation parameter a, corresponding to “linear” amplitude. We find the convergence to be directly affected by an infinite number of movable singularities in the complex a-plane. The singularities occur at locations in the complex plane where the differential equation under consideration is singular. These singularities explain the nonuniform convergence of the perturbation series that Melvin [SIAM J. Appi. Math., 33 (1977), pp. 161–194] noted. They occur in complex conjugate pairs and move in the complex plane as a function of the independent variable, causing divergence of the perturbation series solution. Our analysis near one of these singularities indicates that distinct branches of solutions occur there. These solutions undergo transition at the singularity to develop into new solution branches with distinctively different properties. Moving singularities similar to those discussed here were also shown to affect the convergence of the perturbation expansion for the limit cycle of van der Pol's equation [SIAM J. Appi. Math., 44 (1984), pp. 881–895; SIAM J. Appi. Math., 50 (1990), pp. 1764–1779]. The recurrence of this phenomenon in the present simpler problem suggests that it is very likely to affect the convergence of perturbation expansions for periodic solutions of other problems.