  We present an analytical model for the non-spherical collapse of overdense
regions out of a Gaussian random field of initial cosmological perturbations.
The collapsing region is treated as an ellipsoid of constant density, acted
upon by the quadrupole tidal shear from the surrounding matter. The dynamics of
the ellipsoid is set by the ellipsoid self-gravity and the external quadrupole
shear. Both forces are linear in the coordinates and therefore maintain
homogeneity of the ellipsoid at all times. The amplitude of the external shear
is evolved into the non-linear regime in thin spherical shells that are allowed
to move only radially according to the mass interior to them. We describe how
the initial conditions can be drawn in the appropriate correlated way from a
random field of initial density perturbations. By considering many random
realizations of the initial conditions, we calculate the distribution of shapes
and angular momenta acquired by objects through the coupling of their
quadrupole moment to the tidal shear. The average value of the spin parameter,
0.04, is found to be only weakly dependent on the system mass, the mean
cosmological density, or the initial power spectrum of perturbations, in
agreement with N-body simulations. For the cold dark matter power spectrum,
most objects evolve from a quasi-spherical initial state to a pancake or
filament and then to complete virialization. Low-spin objects tend to be more
spherical. The evolution history of shapes is primarily induced by the external
shear and not by the initial triaxiality of the objects. The statistical
distribution of the triaxial shapes of collapsing regions can be used to test
cosmological models against galaxy surveys on large scales.
