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

TL;DR

This paper proves that the $C^{8/3}$ regularity of simultaneous binary collisions in the collinear 4-body problem extends to the planar case, using advanced geometric methods to analyze the dynamics.

Contribution

It demonstrates that the $C^{8/3}$ differentiability of the block map persists for SBC in the planar 4-body problem, extending previous collinear results.

Findings

$C^{8/3}$ regularity persists in the planar case

Uses geometric tools like blow-up and Dulac maps

Extends regularity results from collinear to planar configurations

Abstract

The dynamics of the 4-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear 4-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only $C^{8/3}$ differentiable. In this paper, it is proved that the $C^{8/3}$ differentiability persists for the SBC in the planar 4-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.