How is Noether's theorem applied in fluid dynamics? As the title says, I'm trying to understand the applications of Noether's theorem in fluid dynamics. I was looking for references in this context, but I only find very old papers, where explanations are very limited. Are there any recent studies for the application of symmetries and conservation laws (momentum, energy conservation ..), for example, in fluid simulators?
 A: You can find a nice discussion about the virtual (added) mass force that arises when a submerged body accelerates relative to the its surrounding fluid in Gregory Falkovich’s Fluid Mechanics book. Apparently, this force is a consequence of the conservation of quasimomentum, which is the conserved quantity associated to the symmetry of the fluid equations with respect to linear translations of the particle in an infinite fluid, while keeping the fluid (bulk) still. This is not a systematic study of the application of Noether’s theorem in the context of fluid mechanics, but I think you will still find this application very relevant.
A: The situation is a little more complicated with continuous systems such as fluids, which are described by both scalar and vector fields. But it's definitely possible.
I'll give a brief, simple example. For continuous systems, a Lagrangian $\mathscr{L}$ is defined using a Lagrangian density function $L$: $$ \mathscr{L} = \int dt \int L \, d^3x.$$
Here, $L = T - U$ just as with point particles. Suppose we have no external potential $U$, but the fluid maintains an internal energy $e$, which typically encodes thermodynamical information about the fluid. So $L$ in this context becomes
$$L=\sum_{i=1}^{3} \frac{1}{2}\rho\left(\frac{\partial x_{i}}{\partial t}\right)^{2}-\rho e$$
(where $e$ generally depends on entropy and the specific volume of the fluid.) If we maintain that our Lagrangian density remains invariant under the transformation $$t \rightarrow t + \delta t$$ then Noether's theorem (in a continuum) reduces down to
$$ \frac{d}{d t} \int \delta t \left[L -\frac{\partial L}{\partial\left(\partial x_{i} / \partial t\right)}\left(\frac{\partial x_{i}}{\partial t} \right)\right] \rho d^3x =0.$$
Using $L$ from above, this is
$$\frac{d}{d t} \int \rho \left[ \frac{1}{2} \sum_{i=1}^{3}\left(\frac{\partial x_{i}}{\partial t}\right)^{2}-e-\frac{1}{2} \sum_{i=1}^{3}\left[2\left(\frac{\partial x_{i}}{\partial t}\right)^{2}\right] \right] d^3 x  = 0,$$
or,
$$\frac{d}{d t} \int \rho\left[\frac{1}{2} \sum_{i=1}^{3}\left(\frac{\partial x_{i}}{\partial t}\right)^{2}+e\right] d^3x=0.$$
This is (total) energy conservation, derived via time-translational invariance.
There are other, non-trivial conservation conditions which may be derived from Noether's theorem. In particular, and I will not derive this here, if you label each infinitesimally small parcel of fluid with some label $a_i$, where $i \in \left\{1,2,3\right\} $, and mandate that $\mathscr{L}$ is invariant under $$a_i \rightarrow a_i + \delta a_i$$ (that is, $\mathscr{L}$ satisfies a particle relabelling symmetry) then you can derive Kelvin's circulation theorem.
If you'd like to read more about this, I highly suggest this (horribly expensive) text from Badin & Crisciani.
