The Classical N-body Problem in the Context of Curved Space

Abstract

AbstractWe provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature κ, for all κ ∊ ℝ. In previous studies, the equations of motion made sense only for κ ≠ 0. The system derived here does more than just include the Euclidean case in the limit κ → 0; it recovers the classical equations for κ = 0. This new expression of the laws of motion allows the study of the N-body problem in the context of constant curvature spaces and thus oòers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals.