The two-body problem with generalized Lennard-Jones potential

We consider a generalization of the famous Lennard-Jones potential. To study the two-body problem associated to this potential, we use the foliations of the phase space by the invariant sets corresponding to the first integrals of energy and angular momentum. We investigate all possible situations created by the interplay among the constants of integration and the field parameters. We obtain the global flow, and illustrate it in both 3D and 2D. This flow exhibits a great variety of orbits, a homoclinic one included. All phase portraits are interpreted in terms of physical trajectories.