  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.
