On co-orbital quasi-periodic motion in the three-body problem

Within the framework of the planar three-body problem we establish the existence of quasi-periodic motions and KAM $4$-tori related to the co-orbital motion of two small moons about a large planet where the moons move in nearly circular orbits with almost equal radii. The approach is based on a combination of normal form and symplectic reduction theories and the application of a KAM theorem for high-order degenerate systems. To accomplish our results we need to expand the Hamiltonian of the three-body problem as a perturbation of two uncoupled Kepler problems. This approximation is valid in the region of phase space where co-orbital solutions occur.Peer ReviewedPostprint (author's final draft

Qualitative features of Hamiltonian systems through averaging and reduction

In this work we analyze the existence and stability of periodic solutions to a Hamiltonian vector field which is a small perturbation of a vector field tangent to the fibers of a circle bundle. By averaging the perturbation over the fibers of the circle bundle one obtains a Hamiltonian system on the reduced (orbit) space of the circle bundle. First we state results which have hypotheses on the reduced system and have conclusions about the full system. The second part is devoted to the application of the general results to the spatial lunar problem of celestial mechanics, i.e. the restricted three-body problem where the infinitesimal is close to one of the primaries. After scaling, the lunar problem is a perturbation of the Kepler problem, which after regularization is a circle bundle flow. We prove the existence of four families of periodic solutions for any small regular perturbation of the spatial Kepler problem: we find the classical near circular periodic solutions and the near rectilinear periodic solutions for all values of the small parameter. Finally we compute their approximate multipliers.Ministerio de Educación y CienciaDepartamento de Educación y Cultura (Gobierno de Navarra