  A reconstruction method for recovering the initial conditions of the Universe
starting from the present galaxy distribution is presented which guarantees
uniqueness of solutions. We show how our method can be used to obtain the
peculiar velocities of a large number of galaxies, hence trace galaxies orbits
back in time and obtain the entire past dynamical history of the Universe above
scales where multi-streaming has not occurred. When tested against a 128^3
LambdaCDM simulation in a box of 200 Mpc/h length, we obtain 60% exact
reconstruction on scales above 6 Mpc/h. We apply our method to a real galaxy
redshift catalogue, the updated NBG (Nearby Galaxies), containing 1483
galaxies, groups and clusters in a radius of 30 Mpc/h, and reconstruct the
peculiar velocity fields in the local neighbourhood. Our reconstructed
distances are well-matched to the observed values outside the collapsed regions
if Omega_m(t)= 0.20 exp(-0.26 (t-13)) where t is the age of the Universe in
Gyrs.
