On the Symmetric Collinear Four-Body Problem

Abstract

The global geometry of the phase structure in a special case of the general Newtonian four-body problem was studied both analytically and numerically in the case of negative energy. Our method consists of blow-up of total collision by McGehee’s coordinates. and representation of orbits by symbol sequences. The analytical study for arbitrary masses clarifies the macroscopic behavior in phase space: escape motion, vortical motion around vertical lines along which the escape motion occurs, and vertical convective flow. We numerically examined a distribution of the symbol sequences on a surface of section in the case of equal masses. The result has clarified that there never exist orbits whose symbol sequences contain special words: un-realizable words. On the other hand, the existence of oscillatory motions are shown under a reasonable assumption. We semi-analytically obtained the initial conditions leading to escape using escape criteria established in the present study. Additionally, we established a way to find the fastest capture-escape orbits, the ejection-collision orbits besides the homothetic solution, the capture-collision orbits, and the ejection-escape orbits. Moreover, quasi-periodic orbits containing a Schubart-like orbit, and unstable periodic orbits were found. The result displays a similarity between the symmetric collinear four-body problem and the collinear three-body problem.