R3 fluids
  With regard to large-scale astrophysical systems, the current paper deals with (i) formulation of tensor virial equations from the standpoint of analytical mechanics; (ii) investigation on the role of systematic and random motions for virial equilibrium configurations; (iii) extent to which systematic and random motions are equivalent in changing a fluid shape. The tensor virial equations are formulated using analytical mechanics, and the self potential-energy tensor is shown to be symmetric. The role of systematic and random motions in collisionless, ideal, self-gravitating fluids, is analysed in detail including radial and tangential velocity dispersion on the equatorial plane. R3 fluids are defined as ideal, self-gravitating fluids in virial equilibrium, with systematic rotation around a principal axis of inertia, and ihe related virial equations are formulated. A unified theory of systematic and random motions is developed for R3 fluids, taking into consideration imaginary rotation. The effect of random motion excess is shown to be equivalent to an additional real or imaginary rotation, respectively, inducing flattening or elongation. R3 fluids are found to admit adjoint configurations with isotropic random velocity distribution. Further constraints are established on the amount of random velocity anisotropy along the principal axes, for triaxial configurations. A necessary condition is formulated for the occurrence of bifurcation points from axisymmetric to triaxial configurations in virial equilibrium, which is independent of the anisotropy parameters. In the special case of homeoidally striated Jacobi ellipsoid, some previously known results are reproduced. 
R. Caimmi