Global phase portraits of the planar perpendicular problem of two fixed centers

Abstract

We study the global phase portrait of the classical problem of an electron in the electrostatic field of two protons that we assume fixed to symmetric distances on the x3 axis. The general problem can be formulated as an integrable Hamiltonian system of three degrees of freedom, but we restrict our study to the invariant planar case that is equidistant to the two fixed centers. This is a two degrees of freedom problem with two constants of motion, the energy and the angular momentum, denoted by H and C, respectively, which are independent and in involution. We describe the foliation of the four-dimensional phase space by the invariant sets of constant energy Ih and we characterize their topology. We also describe the foliation of each energy level Ih by the invariant sets Ihc, and we classify the topology of Ihc and the flow on these invariant sets. In this way we provide a global qualitative description of the motion. We also compare our results with the existing published results.