Second-kind symmetric periodic orbits for planar perturbed Kepler problems and applications (Q6541319)

The authors, here, investigated the existence of families of symmetric periodic solutions of second-kind as continuation of the elliptical orbits of the two-dimensional Kepler problem under of certain symmetric differentiable perturbations using Delaunay coordinates. They characterized the sufficient conditions (for the existence and type of stability) for the continuation of symmetric Keplerian solutions to the complete perturbed problem from the point of view of averaging theory. The results are valid only for some values of the perturbed parameter. The main contribution of this paper is to provide sufficient conditions for the existence of families of second-kind symmetric periodic solutions for planar Keplerian perturbations in inertial frame using Delaunay variables, and they give information about the stability of these solutions. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In continuation, they presented some results about the relationship between their symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. Applications of their work, they get new families of periodic solutions: 1. For the perturbed hydrogen atom with stark and quadratic Zeeman effect, 2. For the anisotropic Seeligers two-body problem, and 3. For the planar generalized Stormer problem.