  Galaxy velocities in clusters, rotation curves of galaxies, and "vertical"
oscillations in the Milky Way currently show too high velocities with respect
to the masses thought to be involved. While these velocity excesses are
currently interpreted as the consequence of dark matter, it can also be
naturally explained as a consequence of scale invariant theory, which rests on
a very simple first principle: the addition of a new fundamental symmetry. In
the present work, the case of scale invariance, present in General Relativity
and Maxwell equations for the empty space without charge and current, is
considered. Cosmological models predict a rapid decrease of these effects with
increasing mean density up to the critical density, where they totally
disappear. Starting from the scale invariant geodesic equation by Dirac (1973),
for which a demonstration by an action principle is presented, a modified
Newton equation is derived. The solutions of this equation are applied to
clusters of galaxies, galactic rotation at different redshifts and "vertical"
motions in the Milky Way. In this new framework, the convergence of theoretical
predictions and observations, in different gravitational systems, epochs, mass
range and spatial scales, opens interesting perspectives that deserve to be
explored further.
