  Luminous Compact Galaxies are enigmatic sources by many aspects. They can
reach the luminosity of the Milky Way within a radius of only a few kpc. They
also represent one of the most rapidly evolving populations of galaxies since
they represent up to 1/5 of the luminous galaxies at redshift z= 0.7 while
being almost absent in the local Universe. The measurement of their dynamics is
crucial to our understanding of LCGs since this has the potential of telling us
which physical process(es) that drives them, and ultimately to link them to the
existing present-day galaxies. Here we derive the 3 dimensional velocity fields
and velocity dispersion (sigma) maps of 17 Luminous Compact Galaxies selected
from the Canada France Redshift Survey and the Hubble Deep Field South with
redshifts ranging from z=0.4 to z=0.75. We find that only 18% of them show
rotational velocity fields typical of rotating disks, the others showing more
complex kinematics. Assuming that LCGs are not too far from equilibrium, about
half of LCGs then appear to be either non-relaxed objects, or objects that are
not supported by velocity dispersion alone. This supports the view that an
important fraction of LCGs are probably mergers. It brings additional support
to the ``spiral rebuilding scenario'' in which LCGs correspond to a previous or
post-merger phase before the disk re-building.
