Example

Show that the Euler-Lagrange field equation (ENT2) for the Lagrange density ${\cal L}(u,x,v)=\tfrac12\norm v^2+\rho(x)u$ is Poisson's equation $\sum \partial_{x_j}^2\psi = \rho$.

For $j=1,\dots, n$ we have $\partial_{v_j}{\cal L}(\psi,.,d\psi) = \tfrac12\cdot 2\cdot v_j|_{v=d\psi} = \partial_{x_j}\psi$. Therefore, the left hand side of (ENT2) is $$ \sum_{j=1}^{n} \partial_{x_j}\left(\partial_{v_j}{\cal L}(\psi,.,d\psi)\right) = \sum_{j=1}^{n}\partial_{x_j}\left(\partial_{x_j}\psi\right)=\sum_{j=1}^{n}\partial_{x_j}^2\psi. $$ On the right hand side of (ENT2), we have $\partial_u{\cal L}(\psi,.,d\psi) = \rho$. So the Euler-Lagrange field equation for the given Lagrange density is indeed Poisson's equation $\sum_{j=1}^{n}\partial_{x_j}^2\psi = \rho$.