13.1 Lagrangian perturbation theory
Following [44
, 45
], 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 [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
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.