Let us consider the simplest case, namely a barotropic ordinary fluid for which . Then we want to perturb the continuity and Euler equations from the previous Section 12. The conservation of mass for the perturbations follows immediately from the Newtonian limits of Equations (134) and (138) (which as we recall automatically satisfy the continuity equation):
We want to rewrite this equation in terms of the displacement vector . After some algebra one finds.
Having derived the perturbed Euler equations, we are interested in constructing conserved quantities that can be used to assess the stability of the system. To do this, we first multiply Equation (237) by the number density , and then write the result (schematically) ascanonical energy of the system as . In the case of an axisymmetric system, e.g. a rotating star, we can also define a canonical angular momentum as
As discussed in [44, 45], the stability analysis is complicated by the presence of so-called “trivial” displacements. These trivials can be thought of as representing a relabeling of the physical fluid elements. A trivial displacement leaves the physical quantities unchanged, i.e. is such that . This means that we must have put it, the “initial relabeling is carried along with the unperturbed motion”.
The trivials have the potential to cause trouble because they affect the canonical energy. Before one can use the canonical energy to assess the stability of a rotating configuration one must deal with this “gauge problem”. To do this one should ensure that the displacement vector is orthogonal to all trivials. A prescription for this is provided by Friedman and Schutz . In particular, they show that the required canonical perturbations preserve the vorticity of the individual fluid elements. Most importantly, one can also prove that a normal mode solution is orthogonal to the trivials. Thus, normal mode solutions can serve as canonical initial data, and be used to assess stability.
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