I wonder that could we write an elliptic PDE on Riemannian manifold with a gradient term $(\theta, d u)_g$ as the E-L equation of a functional, here $\theta$ is a $1-form$ and $(\theta, d u)_g$ is the inner product induced by the metric $g$.
I recall this answer here, which said that the energy functional of $$ -\Delta u+D \varphi \cdot D u=f $$ is $$ I[w]:=\int_U L(D w, w, x) d x $$ with $L(p, z, x)=e^{-\varphi(x)}\left(\frac{1}{2}|p|^2-f(x) z\right)$.
Can we do similar things to the elliptic PDE with a gradient term $(\theta, d u)_g$, for example $$ -\Delta u+(\theta, d u)_g=f, $$ so that the equation could also be written as the E-L equation of a functional?
I tried like this:
if we write $$\langle\theta, d u\rangle=g^{i j} \theta_i \frac{\partial u}{\partial x^j},$$ does this means that when the metric is diagonal and $g^{ii}\theta_i$ could be written as the derivative of some function, then the PDE I mentioned on manifold could be written as the E-L equation of some functional?
