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

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

2000-03-16

arxiv.org/abs/gr-qc/0003064v1

Class.Quant.Grav. 17 (2000) 1949-1962

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

15 pages, to appear in Classical and Quantum Gravity

Scientific paper

10.1088/0264-9381/17/9/306

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.

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