On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Collinear 4-Body Problem

TL;DR

This paper investigates the $ C^{8/3} $ regularity of flow near simultaneous binary collisions in the collinear 4-body problem, revealing the geometric structure of the collision manifold and the reasons for finite differentiability.

Contribution

The paper provides a new proof of the $ C^{8/3} $ regularity and explicitly computes the transition asymptotics near the collision, linking differentiability to the structure of the collision manifold.

Findings

The collision manifold consists of two normally hyperbolic saddle singularity manifolds connected by heteroclinics.

Finite differentiability at $ 8/3 $ is due to the inability to construct local integrals near the collision.

The $ C^{8/3} $ regularity is independent of initial conditions and masses.

Abstract

The singularity at a simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than $ C^{8/3} $ differentiable with respect to initial conditions. Through a blow-up of the singularity, this loss of differentiability is investigated and a new proof of the $ C^{8/3} $ regularity is provided. In the process, it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics. By utilising recent work on transitions near such objects and their normal forms, an asymptotic series of the transition past the singularity is explicitly computed. It becomes remarkably apparent that the finite differentiability at $ 8/3 $ is due to the inability to construct a set of integrals local to the simultaneous binary collision. The finite differentiability is shown to be independent from a choice of initial condition or value of the masses.