  The Lagrangian dynamics of a single fluid element within a self-gravitational
matter field is intrinsically non-local due to the presence of the tidal force.
This complicates the theoretical investigation of the non-linear evolution of
various cosmic objects, e.g. dark matter halos, in the context of Lagrangian
fluid dynamics, since a fluid parcel with given initial density and shape may
evolve differently depending on their environments. In this paper, we provide a
statistical solution that could decouple this environmental dependence. After
deriving the probability distribution evolution equation of the matter field,
our method produces a set of closed ordinary differential equations whose
solution is uniquely determined by the initial condition of the fluid element.
Mathematically, it corresponds to the projected characteristic curve of the
transport equation of the density-weighted probability density function (PDF).
Consequently it is guaranteed that the one-point PDF would be preserved by
evolving these local, yet non-linear, curves with the same set of initial data
as the real system. Physically, these trajectories describe the mean evolution
averaged over all environments by substituting the tidal tensor with its
conditional average. For Gaussian distributed dynamical variables, this mean
tidal tensor is simply proportional to the velocity shear tensor, and the
dynamical system would recover the prediction of Zel'dovich approximation (ZA)
with the further assumption of the linearized continuity equation. For Weakly
non-Gaussian field, the averaged tidal tensor could be expanded perturbatively
as a function of all relevant dynamical variables whose coefficients are
determined by the statistics of the field.
