Dynamics in the Charged Restricted Circular Three-Body Problem

Abstract

The existence and stability of periodic solutions for different types of perturbations associated to the Charged Restricted Circular Three Body Problem (shortly, CHRCTBP) is tackled using reduction and averaging theories as well as the technique of continuation of Poincaré for the study of symmetric periodic solutions. The determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the occurrence of Hamiltonian-Hopf bifurcations associated to some equilibrium points of the CHRCTBP. 2 and L coll 3 and the isosceles triangle equilibrium L iso 4 and L iso 5 were characterized

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

“…In celestial mechanics, one of the most well-known integrable model is the Kepler problem. There exist many other problems that are formulated as a perturbation of the Kepler problem in Cartesian coordinates (see [5,6,10,13] and references therein) or in rotating coordinates (see, e.g., [16,21,25,27]). Poincaré [28] considered the investigation of periodic solutions of the restricted three-body problem where, in particular, he classified the periodic orbits of second kind that are generated by the elliptic orbits of the planar Kepler problem (the first kind are generated by the circular orbits of the planar Kepler problem).…”

“…There are considerable works that have contributed to find the periodic solution for perturbations of an integrable problem (see [1,7]). For a perturbed Kepler problem, some of these works apply the method of average of first kind (see, e.g., [3,4,14,25,27]). The technique of combining discrete symmetries of Hamiltonian and the Poincaré continuation method has been considered in other works for twodegree-of-freedom (2-DOF) and three-degree-of-freedom problems (see, e.g., [5,10,25,[31][32][33][34][35]).…”

“…For the first kind of symmetric periodic solutions, we recommend the reader the references [5,10,34,35]. For the second kind of symmetric periodic solutions, we mention [2,8,21,25,33].…”

“…With regard to the study of periodic orbits of the second kind in perturbed Kepler problems, we can cite [2, 25, 31] and references therein. There are standard results in celestial mechanics which state that every elliptical solution of period ( and are relative prime positive integers) of the planar rotating Kepler problem with initial condition on the -axis can be extended to the perturbed planar circular restricted -body problem.…”

“…There are a significant number of authors who search for periodic solutions to perturbations of the planar Kepler problem (in inertial frame) using Delaunay coordinates, but the technique used is the average method. Important references in this regard can be found in [2, 3, 20, 25, 27]. Many other authors considered the study of symmetric periodic orbits of first or second kind for planar perturbations of the Kepler problem.…”