How to get Euler-Lagrange equations out? Find the Euler-Lagrange equations associated for 
$$ \int dt \int d^3x \left\{ \frac12\rho|\mathbf{v}|^2 - u(\rho) + \phi [\dot\rho + \nabla \cdot (\rho \mathbf{v})] \right\}$$
where $\rho =\rho(t, \mathbf{x})$, $\mathbf{v} = \mathbf{v}(t, \mathbf{x})$, $u = u(\rho)$ and $\phi =\phi(t, \mathbf{x}) $ is a Lagrange multiplier.
Struggling since $\mathbf{\dot x} \neq \mathbf{v}$ (it is a velocity field), and not too sure what variables to differentiate with respect to. 
 A: I'll expand on @QMechanic's answer. The first equation is straightforward. For the second, note your action is$$S=\int d^4x\mathcal{L},\,\mathcal{L}:=\frac12\rho v_iv_i-u(\rho)+\phi\dot{\rho}+\phi\rho\partial_iv_i+\phi v_i\partial_i\rho,$$so$$0=\frac{\partial\mathcal{L}}{\partial v_i}-\partial_j\frac{\partial\mathcal{L}}{\partial\partial_jv_i}=\rho v_i+\phi \partial_i\rho-\partial_j\left(\phi\rho\delta_{ij}\right)=\rho v_i-\rho\partial_i\phi=\rho\left(v-\nabla\phi\right)_i,$$which reduces to the given result. Finally, varying $\rho$ gives$$0=\frac{\partial\mathcal{L}}{\partial\rho}-\partial_j\frac{\partial\mathcal{L}}{\partial\partial_j\rho}-\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\rho}}=\frac12 v^2-u^\prime+\phi\nabla\cdot v-\partial_j\left(\phi v_j\right)-\dot{\phi}\\=\frac12 v^2-u^\prime-v\nabla\phi-\dot{\phi},$$which again is what was claimed.
Incidentally, a physical interpretation of these equations provides a sanity check: @QMechanic's first result was mass conservation, the second identified $v$ with $\phi$'s gradient as we expected, and third is Navier-Stokes and analogous to Newton's second law.
A: The EL equations become:


*

*Variation wrt. $\phi$ $\qquad\Rightarrow \qquad \dot{\rho} + \nabla \cdot (\rho \mathbf{v})~=~0.$

*Variation wrt. $\mathbf{v}$ $\qquad\Rightarrow \qquad \mathbf{v}~=~ \nabla\phi .$

*Variation wrt. $\rho$ $\qquad\Rightarrow \qquad \frac{1}{2}|\mathbf{v}|^2 ~=~ u^{\prime}(\rho) + \dot{\phi} + \mathbf{v}\cdot \nabla\phi  .$
