  This paper focusses on the barely understood gap between the weakly nonlinear
regime of structure formation and the onset of the virialized regime. While the
former is accessed through perturbative calculations and the latter through
virialization conditions incorporating dynamical stresses that arise in
collisionless self-gravitating systems due to velocity dispersion forces, the
addressed regime can only be understood through non-perturbative models. We
here present an exact Lagrangian integral that provides a tool to access this
regime. We derive a transport equation for the peculiar-gravitational field
strength and integrate it along comoving trajectories of fluid elements. The
so-obtained integral provides an exact expression that solves the longitudinal
gravitational field equation in general. We argue that this integral provides a
powerful approximation beyond the Lagrangian perturbative regime, and discuss
its relation to known approximations, among them Lagrangian perturbation
solutions including the Zel'dovich approximation and approximations for
adhesive gravitational clustering, including the adhesion approximation.
Furthermore, we propose an iteration scheme for a systematic analytical and
numerical construction of trajectory fields. The integral may also be employed
to improve inverse reconstruction techniques.
