The non-existence of centre-of-mass and linear-momentum integrals in the curved N-body problem

Abstract

Abstract. We provide a class of orbits in the curved N-body prob-lem for which no point that could play the role of the centre of mass isfixed or moves uniformly along a geodesic. This proves that the equa-tions of motion lack centre-of-mass and linear-momentum integrals. 1. IntroductionThe importance of the centre-of-mass and linear-momentum integrals inthe classical N-body problem and its generalization to quasihomogeneouspotentials, [3], is never enough emphasized. From integrable systems, tothe Smale and Saari conjectures, [30], [37], to the study of singularities,[4], [10], these integrals have played an essential role towards answeringthe most basic questions of particle dynamics, in general, and celestialmechanics, inparticular. But theyseem tocharacterize onlythe Euclideanspace. Away from zero curvature, they disappear, probably because of thesymmetry loss that occurs in the ambient space. We will show here thatthey don’t exist in the curved N-body problem (the natural extension ofthe Newtonian N-body problem to spaces of constant curvature).This note follows from a discussion we had with Ernesto P´erez-Chavelaand Guadalupe Reyes Victoria of Mexico City. Given the absence of theabove mentioned integrals in discretizations of Einstein’s field equations,such as those of Levi-Civita, [23], [24], Einstein, Infeld, Hoffmann, [13],and Fock, [14], we had taken this property for granted in the curved N-body problem. But as we could not immediately support this claim with