  We write the partition function for a lattice gauge theory, with compact
gauge group, exactly in terms of unconstrained variables and show that, in the
mean field approximation, the dynamics of pure gauge theories, invariant under
compact, continuous,groups of rank 1 is the same for all. We explicitly obtain
the equivalence for the case of SU(2) and U(1) and show that it obtains, also,
if we consider saddle point configurations that are not,necessarily, uniform,
but only proportional to the identity for both groups. This implies that the
phase diagrams of the (an)isotropic SU(2) theory and the (an)isotropic U(1)
theory in any dimension are identical, within this approximation, up to a
re-evaluation of the numerical values of the coupling constants at the
transitions. Only nonuniform field configurations, that, also, belong to higher
dimensional representations for Yang--Mills fields, will be able to p robe the
difference between them. We also show under what conditions the global symmetry
of an anisotropic term in the lattice action can be promoted to a gauge
symmetry of the theory on layers and point out how deconstruction and flux
compactification scenaria may thus be studied on the lattice.
