  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.
