The Newtonian n-Body Problem in the Context of Curved Space

Abstract

The idea that geometry and physics are intimately related made its way in human thought during the early part of the nineteenth century. Gauss measured the angles of a triangle formed by three mountain peaks near Göttingen, Germany, apparently hoping to learn whether the universe has positive or negative curvature, but the inevitable observational errors rendered his results inconclusive [3]. In the 1830s, Bolyai and Lobachevsky took these investigations further. They independently addressed the connection between geometry and physics by seeking a natural extension of the gravitational law from Euclidean to hyperbolic space. Their idea led to the study of the Kepler problem and the 2-body problem in spaces of nonzero constant Gaussian curvature, κ ≠ 0, two fundamental problems that are not equivalent, unlike in Euclidean space. A detailed history of the results obtained in this direction since Bolyai and Lobachevsky can be found in [3, 5, 6].