Energy principles for self-gravitating barotropic flows - II. The stability of Maclaurin discs

We analyse the stability conditions of Maclaurin discs 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. We find the necessary and sufficient conditions of stability for single-mode, symmetrical or antisymmetrical, perturbations. The limits of stability are identical with those given by dynamical analysis. Application of our constrained energy-minimum principle to Maclaurin discs thus suggests that it may be a powerful tool, next to dynamical perturbations, to find stability limits in other galactic models.