Planar symmetric periodic orbits in four dipole problem

We have extend Stormer's problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme "orderlines" such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn. For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions we investigate these motions there is enormous three dimensional instability. Following the Poincare's hypothesis that periodic solutions are `dense' in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families of three dimensional symmetric periodic orbits.