Stability in the Weak Variational Principle of Barotropic Flows

I find conditions under which the ”Weak Energy Principle” of Katz, Inagaki and Yahalom (1993) gives necessary and sufficient conditions. My conclusion is that, necessary and sufficient conditions of stability are obtained when we have only two mode coupling in the gyroscopic terms of the perturbed Lagrangian. To illustrate the power of this new energy principle, I have calculated the stability limits of two dimensional configurations such as ordinary Maclaurin disk, an infinite self gravitating rotating sheet, and a two dimensional Rayleigh flow which has well known sufficient conditions of stability. All perturbations considered are in the same plane as the configurations. The limits of stability are identical with those given by a dynamical analysis when available, and with the results of the strong energy principle analysis when given. Thus although the ”Weak Energy” method is mathematically more simple than the ”Strong Energy” method of Katz, Inagaki and Yahalom )1993) since it does not involve solving second order partial differential equations, it is by no means less effective.