Noncollision Singularities in a Planar Four-body Problem

TL;DR

This paper proves the Painlevé conjecture for the four-body problem by demonstrating a set of initial conditions leading to finite-time escape without collisions, confirming a long-standing open problem.

Contribution

It establishes the existence of noncollision singularities in the planar four-body problem, settling the last open case of the Painlevé conjecture.

Findings

Existence of a Cantor set of initial conditions with finite-time escape

Proof of noncollision singularities in the four-body problem

Resolution of the Painlevé conjecture for four bodies

Abstract

In this paper, we show that there is a Cantor set of initial conditions in the planar four-body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlev\'{e} conjecture for the four-body case, and thus settles the last open case of the conjecture.