### 13.1 Lagrangian perturbation theory

Following [4445], we work with Lagrangian variations. We have already seen that the Lagrangian perturbation of a quantity is related to the Eulerian variation by
where (as before) is the Lie derivative that was introduced in Section 2. The Lagrangian change in the fluid velocity follows from the Newtonian limit of Equation (135):
where is the Lagrangian displacement. Given this, and
we have

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):

Consequently, the perturbed gravitational potential follows from
In order to perturb the Euler equations we first rewrite Equation (218) as
where . This form is particularly useful since the Lagrangian variation commutes with the operator . Perturbing Equation (233) we thus have

We want to rewrite this equation in terms of the displacement vector . After some algebra one finds

Finally, we need
Given this, we have arrived at the following form for the perturbed Euler equation:
This equation should be compared to Equation (15) of [44].

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) as

omitting the indices since there is little risk of confusion. Defining the inner product
where and both solve the perturbed Euler equation, and the asterisk denotes complex conjugation, one can now show that
The latter requires the background relation , and holds provided that at the surface of the star. A slightly more involved calculation leads to
Inspired by the fact that the momentum conjugate to is , we now consider the symplectic structure
Given this, it is straightforward to show that is conserved, i.e. . This leads us to define the canonical energy of the system as
After some manipulations, we arrive at the following explicit expression:
which can be compared to Equation (45) of [44]. In the case of an axisymmetric system, e.g. a rotating star, we can also define a canonical angular momentum as
The proof that this quantity is conserved relies on the fact that (i) is conserved for any two solutions to the perturbed Euler equations, and (ii) commutes with in axisymmetry, which means that if solves the Euler equations then so does .

As discussed in [4445], 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

The solution to the first of these equations can be written as
where, in order to satisfy the second equations, the vector must have time-dependence such that
This means that the trivial displacement will remain constant along the background fluid trajectories. Or, as Friedman and Schutz [44] 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 [44]. 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.