Investigations into the stability properties of rotating self-gravitating bodies are of obvious relevance to astrophysics. By improving our understanding of the relevant issues we can hope to shed light on the nature of the various dynamical and secular instabilities that may govern the spin-evolution of rotating stars. The relevance of such knowledge for neutron star astrophysics may be highly significant, especially since instabilities may lead to detectable gravitational-wave signals. In this section we will review the Lagrangian perturbation framework developed by Friedman and Schutz [44, 45] for rotating non-relativistic stars. This will lead to criteria that can be used to decide when the oscillations of a rotating neutron star are unstable. We provide an explicit example proving the instability of the so-called r-modes at all rotation rates in a perfect fluid star.
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