2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/95350The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured that the converse is also true for the planar Newtonian N-body problem: relative equilibria are the only constant-inertia solutions. A computer-assisted proof for the 3-body case was recently given by R. Moeckel. We present a different kind of answer: proofs that several generalisations of Saari's conjecture are generically true. Our main tool is jet transversality, including a new version suitable for the study of generic potential functions.21 pagesDynamical SystemsDifferential GeometryGeometric TopologyMSC-class: 70F10 (Primary); 57N75, 37J05 (Secondary)text