We formulate the problem of an infinitesimal particle moving in the space under the influence of the gravitational force induced by a homogeneous annulus disk fixed on a plane. We compute using different coordinates and in terms of an elliptic integral the potential associated to this problem. Also we study the symmetries of the associated potential. After that, we look at the dynamics in some particular cases, namely: in the line perpendicular to the plane that contains the annulus disk and passes through the center of the ring and on the plane that contains the massive annulus.
We investigate the existence of several families of symmetric periodic solutions as continuation of circular orbits of the Kepler problem for certain symmetric differentiable perturbations using an appropriate set of Poincaré-Delaunay coordinates which are essential in our approach. More precisely, we try separately two situations in an independent way, namely, when the unperturbed part corresponds to a Kepler problem in inertial cartesian coordinates and when it corresponds to a Kepler problem in rotating coordinates on R 3 . Moreover, the characteristic multipliers of the symmetric periodic solutions are characterized. The planar case arises as a particular case. Finally, we apply these results to study the existence and stability of periodic orbits of the Matese-Whitman Hamiltonian and the generalized Størmer model.
Applying perturbation theory and symmetry conditions, we prove the existence of new families of periodic orbits for the 3-dimensional anisotropic Manev problem, which depends on three parameters.
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