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Abstract

A variational principle is constructed for the eigenfrequencies of the normal modes of perfect fluid stars which are axisymmetric, differentially rotating, but without meridional circulation. It is the first such principle to be capable of yielding the complex as well as the real eigenfrequencies. The key to the principle is the discovery of a simple way of turning a right eigenvector for a normal mode into a left one. This is based on a symmetry of the unperturbed star, that it is invariant under reflection in time and azimuth. The variational expression also gives a new approach to approximating the linear perturbation problem by one restricted to a finite-dimensional set of trial functions. In cases where it differs from the more conventional approach, the new method should be more accurate.