This dissertation contains work on the simultaneous binary collision in the n-body problem. Martínez and Simó have conjectured that removal of the singularity at this collision via block regularisation results in a regularised flow that is no more than C^(8/3) differentiable with respect to initial conditions. Remarkably, the same authors proved the conjecture for the collinear 4-body problem. The conjecture remains open for the planar case or for n > 4 . This thesis explores the loss of differentiability in the collinear and planar 4-body problem.
In the collinear problem, a new proof is provided of the C^(8/3)-regularisation. In the planar problem, a proof that the simultaneous binary collisions are at least C^2-regularisable is given. In both cases a remarkable link between the finite differentiability and the inability to construct a set of integrals local to the singularities is established. The theoretical framework for improving the C^2 result in the plane is established.
The method of proof in both cases brings together the theory of blow-up, normal forms, hyperbolic transitions, and computation of regular transition maps to explicitly compute an asymptotic expansion of the transition past the singularities. These tools are first explored in novel work on the regularisation of a generic class of degenerate singularities in planar vector fields. In particular, a relatively simple perturbation of an example derived from the 4-body problem is shown to be C^(4/3)
However, the study of simultaneous binary collisions requires that each of these tools be extended to higher dimensions, in particular to manifolds of normally hyperbolic fixed points. General theory on normal forms and asymptotic properties of nearby transitions of such manifolds are detailed. The normal forms are studied in the formal and C^k categories. The hyperbolic transitions are shown to have similar properties to the well studied Dulac maps of planar saddles.