Existence and linear stability of symmetric periodic orbits in the generalized planar Stark-Zeeman problem (Q6572426)

The article investigates the Hamiltonian system of a hydrogen atom in the plane subject to external magnetic and electric fields in the case where the external fields are weak. Thus, the considered Hamiltonian is regarded as a perturbation to a Kepler system.\N\NThe author establishes results concerning the existence and linear stability of so called \(S\)-symmetric periodic solutions to the perturbed Hamiltonian which are near-elliptic, that is, close to elliptic solutions of the Kepler system. Here, \(S\) is the time reversible symmetry with respect to the \(x\)-axis. The established results are valid in certain regions of the parameters of the investigated Hamiltonian.\N\NThe article considers in particular the case where the electric field is absent. In this case, the existence and linear stability of near-circular solutions is established, their characteristic multipliers are identified, and the existence of KAM \(2\)-tori around them is shown.