  In this paper we present a new local Lagrangian approximation to the
gravitational dynamics of cold matter. We describe the dynamics of a Lagrangian
fluid element through only one quantity, the deformation tensor. We show that
this tensor is clearly suited to the study of gravitational dynamics and,
moreover, that knowing its evolution is enough to completely describe a fluid
element. Our approximate dynamics treat exactly the conservation of mass, of
the velocity divergence and of the shear and is exact in the case of planar,
cylindrical and spherical collapses. It also reproduces very accurately the
evolution of all dynamical quantities for a very wide class of initial
conditions as illustrated by a detailed comparison with the homogeneous
ellipsoid model. Beside, we highlight, for the first time, the important
dynamical role played by the Newtonian counterpart of the magnetic part of the
Weyl tensor.
