Variational Derivation of Euler's Equation for Inviscid Flow

The mass density $\rho(t,\mathbf{x})$ and velocity field $\mathbf{v}(t,\mathbf{x})$ of a compressible fluid are constrained by conservation of mass to satisfy the continuity equation \[\dot{\rho} + \nabla\cdot(\rho\mathbf{v}) = 0\,. \tag{1.1}\] Given that the energy density of the fluid is $u(\rho)$, the action (for inviscid irrotational flow) is \[S[\rho,\mathbf{v},\phi] = \int d t\int d^3x\left\{ \tfrac{1}{2}\rho|\mathbf{v}|^2 - u(\rho) + \phi\left[\dot{\rho}+\nabla\cdot(\rho\mathbf{v})\right] \right\},\tag{1.2}\] where $\phi(t,\mathbf{x})$ is a Lagrange multiplier field imposing the continuity condition (1.1). Find the Euler--Lagrange equations for this action. Show that they imply $\mathbf{v}=\nabla\phi$ (so $\phi$ is the velocity potential). Given that the fluid pressure $P(t,\mathbf{x})$ satisfies \[\nabla P = \rho\nabla h(t,\mathbf{x})\,, \qquad h = u'(\rho)\,,\tag{1.3}\] deduce Euler's equation for inviscid irrotational flow: \[\rho\left[\dot{\mathbf{v}} + (\mathbf{v}\cdot\nabla)\mathbf{v}\right] = -\nabla P\,.\tag{1.4}\]