The geometry of the Barbour-Bertotti theories I. The reduction process

TL;DR

This paper explores the geometric structure of the configuration space in Barbour-Bertotti theories for multiple particles, revealing how the reduction process leads to a Riemannian metric linked to physical symmetries and inertia.

Contribution

It introduces a geometric reduction framework for N-particle dynamics in Barbour-Bertotti theories, connecting the metric to the space of orbits and inertia properties.

Findings

The reduced Lagrangian is a Riemannian line element.

The metric is degenerate in flat configurations.

The scalar curvature relates to principal moments of inertia.

Abstract

The dynamics of $N\geq 3$ interacting particles is investigated in the non-relativistic context of the Barbour-Bertotti theories. The reduction process on this constrained system yields a Lagrangian in the form of a Riemannian line element. The involved metric, degenerate in the flat configuration space, is the first fundamental form of the space of orbits of translations and rotations (the Leibniz group). The Riemann tensor and the scalar curvature are computed by a generalized Gauss formula in terms of the vorticity tensors of generators of the rotations. The curvature scalar is further given in terms of the principal moments of inertia of the system. Line configurations are singular for $N\neq 3$. A comparison with similar methods in molecular dynamics is traced.