Algebraic mean field theory constructs a group theoretical model of quantum systems that have a weak dynamical symmetry but may break dynamical symmetry. The strong defining condition for dynamical symmetry is that states belong to one irreducible representation space. Weak dynamical symmetry demands that the densities corresponding to the states have a constant value for each Casimir. Quantum phase transitions and other complex systems exhibit weak dynamical symmetry. Furthermore mean field theory often yields analytic formulae for expectations and energy spectra that are not feasible in representation theory. This paper develops mean field theory on any coadjoint orbit of su(4) densities. The simple Lie algebra su(4) so(6) is a 15-dimensional algebra that contains the subalgebra usp(4) so(5) and the angular momentum algebra su(2). The su(4) dual space consists of density matrices which are defined by the expectations of the su(4) generators. A coadjoint orbit is a common level surface in the dual space of the three su(4) Casimirs. A Lax pair determines the dynamics of these densities on each coadjoint orbit. Analytic solutions are reported for rotating su(4) densities in equilibrium for a particular energy function.