  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.
