  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 of particle positions --- a context in which
subtle spurious effects originating from the singularity in the two body
potential become particularly clear. We focus on the case of spherical systems,
which support only regular orbits in the collisionless limit, despite the
persistence of local exponential instability of $N$-body trajectories in the
anomalous case of discrete point particle representation even as $N \rightarrow
\infty$. When the singularity in the potential is removed, this apparent
contradiction disappears. In the absence of fluctuations, equilibrium
configurations generally correspond to positive scalar curvature, and thus
support stable trajectories. 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, and are necessarily different from the systems dominated by
isothermal cores, expected from entropy maximization in the context of the
standard theory of violent relaxation. It is suggested that this is the case
because a system starting far from equilibrium does not reach a 'most probable
state' via violent relaxation, but that this process comes to an end as the
system finds and (settles in) a configuration where it can most efficiently
wash out perturbations. We explicitly test this interpretation by means of
direct simulations.
