Problem of two fixed centers and a finite dipole: A unified treatment

We investigate the classical dynamics of the problem of two centers and a finite dipole by means of a common Hamiltonian model. Conditions for trapped orbits are determined first by a qualitative analysis of the effective potential, revealing two types of bifurcation in the two-center problem as a control parameter passes through a critical value μc. For equal masses there is a pitchfork bifurcation, for unequal masses a tangent bifurcation. Separating the common Hamiltonian in elliptic coordinates shows that the third invariants for the two-center problem and the finite dipole are isomorphic in scaled variables. Explicit trapping conditions are then found in terms of the coefficients of two quartics. A critical-point analysis for the finite dipole shows that a potential well exists for all values of scaled angular momentum below the same critical value μc, at which the elliptic point runs off to infinity. In this case the existence of the third invariant does not confine any orbits not already trapped by energy conservation. A similar analysis of the effective potential for the point dipole shows that the only trapped orbits besides those impacting the origin are unstable zero-energy trajectories lying on a sphere.