No hyperbolic pants for the 4-body problem with strong potential

Connor Jackman and Richard Montgomery

Vol. 280 (2016), No. 2, 401–410

DOI: 10.2140/pjm.2016.280.401

Abstract

The $N$-body problem with a $1∕r^2$ potential has, in addition to translation and rotational symmetry, an effective scale symmetry which allows its zero energy flow to be reduced to a geodesic flow on complex projective $\left(N-2\right)$-space, minus a hyperplane arrangement. When $N=3$ we get a geodesic flow on the 2-sphere minus three points. If, in addition we assume that the three masses are equal, then it was proved in a previous paper that the corresponding metric is hyperbolic: its Gaussian curvature is negative except at two points. Does the negative curvature property persist for $N=4$, that is, in the equal mass $1∕r^2$ potential 4-body problem? Here we prove that it does not by computing that the corresponding Riemannian metric in this $N=4$ case has positive sectional curvature at some 2-planes. This curvature computation underlines an essential difference between the 3- and 4-body problem, a difference whose consequences remain to be explored.

Keywords

$N$-body problems, dynamical systems, differential geometry, hyperbolic flows

Mathematical Subject Classification 2010

Primary: 37N05, 70F10, 70G45

Milestones

Received: 5 February 2015

Revised: 8 May 2015

Accepted: 3 September 2015

Published: 28 January 2016