  The origin of equilibrium gravitational configurations is sought in terms of
the stability of their trajectories, as described by the curvature of their
Lagrangian configuration manifold. We focus on the case of spherical systems,
which are integrable in the collisionless (mean field) limit despite the
apparent persistence of local instability of trajectories even as $N
\rightarrow \infty$. It is shown that when the singularity in the potential is
removed, a null scalar curvature is associated with an effective, averaged,
equation of state describing dynamically relaxed equilibria with marginally
stable trajectories. The associated configurations are quite similar to those
of observed elliptical galaxies and simulated cosmological halos. This is the
case because a system starting far from equilibrium finally settles in a state
which is integrable when unperturbed, but where it can most efficiently wash
out perturbations. We explicitly test this interpretation by means of direct
simulations.

Cosmology and Nongalactic Astrophysics;Chaotic Dynamics