Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law (from wikipedia https://en.wikipedia.org/wiki/Noether%27s_theorem). Then how is the conservation of mass, momentum and energy in fluids related to Noether's theorem?
For example we can consider the Euler equation and take $X = \{ \rho, \rho u \}$ and the Hamiltonian $H = \int \frac{1}{2}\rho u^2 + V(\rho) $ and one could get the compressible Euler equation using $\dot{X} = \{X,H\}$ where $\{,\}$ is the Poisson bracket. Then how can we apply Noether's theorem to this Hamiltonian?
