Clebsch-potential variational derivation of incompressible Euler flow
= Clebsch-potential variational derivation of incompressible Euler flow
{c}
For $\mathbf u=\nabla\phi+\beta\nabla\alpha$, the density
$$
\mathcal L=-\beta\alpha_t-\frac12|\mathbf u|^2
$$
gives incompressibility and material conservation of $\alpha$ and $\beta$. Differentiating the Clebsch representation then yields the Euler momentum equation with
$$
p=-\frac12|\mathbf u|^2-\phi_t-\beta\alpha_t.
$$