Cristina Stoica - The relative two-body problem in quasi-homogeneous potentials fields

The relative two-body problem in a so-called quasi-homogeneous potential field (i.e. the interaction between the particles is generated by a sum of the form W=U+V, with U and V homogeneous functions in 1/r, r being the distance between the particles) is presented.

Using the McGehee type diffeomorphisms and a suitable reparametrization of time, the initial second order differential system is transformed to an analytic system. The flow is constrained to be on the energy manifold, here a class C-1 relation.

The submanifold r=0, called the collision manifold, is invariant under the flow and the angular momentum conservation isolates the angular velocity component.

We will offer a full description of the orbits on and about the collision manifold, pointing out for each different case with respect to the degrees of U and V the Lebesgue measure of the set of initial conditions leading to collision.