Almost all 3-body relative equilibria on S2 and H2 are inclined

Abstract

We answer here a question posed by F. Diacu in 2012 that asked whether there exist relative equilibria on S2 and H2 that move in a plane non-perpendicular to the rotation axis. For 3-body non-geodesic ordinary central configurations on S2 and H2, we find all relative equilibria that move in a plane perpendicular to the rotation axis. We also show that the set of shapes of 3-body non-geodesic ordinary central configurations on S2 and H2 is a 3-dimensional manifold. Then we conclude that almost all 3-body relative equilibria move in planes non-perpendicular to the rotation axis. © 2019 American Institute of Mathematical Sciences. All rights reserved.