The geometry of the Barbour-Bertotti theories II. The three body problem

TL;DR

This paper develops a geometric framework for analyzing the three-body problem within Barbour-Bertotti theories, exploring curvature properties, symmetries, and geodesic solutions in a non-relativistic setting.

Contribution

It introduces a detailed geometric approach to the three-body problem, including curvature analysis, symmetry exploitation, and integration of geodesic motions in Barbour-Bertotti theories.

Findings

Sectional curvatures of key surfaces are computed.

Geodesic motions are explicitly integrated.

Curvature singularities are analyzed for line configurations.

Abstract

We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for $N\neq 3$, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there.