Take an inviscid, incompressible fluid, ignore external forces for the sake of simplicity.
The Lagrangian density is
$$ \mathcal{L} = \frac{\rho}{2} {\vec v}\cdot \vec v $$
I'm trying to solve Euler-Lagrange like so: $$ \frac{\partial \mathcal{L}}{\partial \vec v} = \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial(\partial_\mu \vec v)} \right) $$
The RHS is $0$ because $\mathcal{L}$ does not depend on any derivatives of $\vec v$.
The LHS is $\rho \vec v$.
This is nonsense, I was expecting to get the momentum equations for an inviscid incompressible fluid.
