Non-Collision singularities in the Planar two-Center-two-Body problem

TL;DR

This paper demonstrates the existence of initial conditions in a simplified four-body system where velocities become infinite in finite time without collisions, supporting the Painlevé conjecture in a planar setting.

Contribution

It proves the existence of non-collision singularities in a simplified planar four-body model, extending understanding of singular behaviors in gravitational systems.

Findings

Existence of initial conditions leading to infinite velocities in finite time.

Construction of a Cantor set of such initial conditions.

Validation of a simplified model related to the Painlevé conjecture.

Abstract

In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers $Q_1=(-\chi,0)$, $Q_2=(0,0)$ of masses 1, and two moving bodies $Q_3$ and $Q_4$ of masses $\mu\ll 1$. They interact via Newtonian potential. $Q_3$ is captured by $Q_2$, and $Q_4$ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlev\'{e} conjecture.