2005/10/01 by Tanya Schmah, Cristina Stoica, Schmah, Tanya +1
Computer Science · Mathematics · #37J05 (Secondary) #57N75 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #MSC-class: 70F10 (Primary) #Matrix Theory and Algorithms #Meromorphic and Entire Functions #math.DG #math.DS #math.GT #msc:37J05 #msc:57N75 #msc:70F10
paper · pdf · doi:10.48550/arxiv.math/0510012
21 pages
arxiv created 2005/10/01 · openalex publication_date 2005/10/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The 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.