Mean field theory of the general collective motion algebra gcm(3)

The Lie algebra gcm(3) is a 15-dimensional semidirect sum that contains the algebra of the general linear group GL+(3, R). The six-dimensional Abelian ideal of gcm(3) is generated by the mass quadrupole and monopole tensors. The Bohr-Mottelson quantum model of nuclear rotational and vibrational states is a particular irreducible representation of gcm(3). The gcm(3) dual space consists of density matrices which are defined by the expectations of the gcm(3) generators. A coadjoint orbit is a level surface in the dual space of the gcm(3) Casimir, which is the squared length of the conserved Kelvin circulation vector. This paper develops mean field theory on any coadjoint orbit of gcm(3) densities. A Lax pair determines the dynamics of gcm(3) densities on each coadjoint orbit. This Lax equation is equivalent to the Riemann ellipsoid equations of motion