  It is well known that all rotating perfect fluid stars in general relativity
are unstable to certain non-axisymmetric perturbations via the
Chandrasekhar-Friedman-Schutz (CFS) instability. However, the mechanism of the
CFS instability requires, in an essential way, the loss of angular momentum by
gravitational radiation and, in many instances, it acts on too long a timescale
to be physically/astrophysically relevant. It is therefore of interest to
examine the stability of rotating, relativistic stars to axisymmetric
perturbations, where the CFS instability does not occur. In this paper, we
provide a Rayleigh-Ritz type variational principle for testing the stability of
perfect fluid stars to axisymmetric perturbations, which generalizes to
axisymmetric perturbations of rotating stars a variational principle given by
Chandrasekhar for spherical perturbations of static, spherical stars. Our
variational principle provides a lower bound to the rate of exponential growth
in the case of instability. The derivation closely parallels the derivation of
a recently obtained variational principle for analyzing the axisymmetric
stability of black holes.
