  We analyze stability conditions of "Maclaurin flows" (self-gravitating,
barotropic, two dimensional, stationary streams moving in closed loops around a
point) by minimizing their energy, subject to fixing all the constants of the
motion including mass and circulations. Necessary and sufficient conditions of
stability are obtained when gyroscopic terms in the perturbed Lagrangian are
zero. To illustrate and check the properties of this new energy principle, we
have calculated the stability limits of an ordinary Maclaurin disk whose
dynamical stability limits are known. Perturbations are in the plane of the
disk. We find all necessary and sufficient conditions of stability for single
mode symmetrical or antisymmetrical perturbations. The limits of stability are
identical with those given by a dynamical analysis. Regarding mixed types of
perturbations the maximally constrained energy principle give for some the
necessary and sufficient condition of stability, for others only sufficient
conditions of stability. The application of the new energy principle to
Maclaurin disks shows the method to be as powerful as the method of dynamical
perturbations.
