I want to extremize an integral via Euler Lagrange equations on a Riemannian manifold. My functional depends on multiple partial derivatives, so the situation that physicists call the "field theoretic" case applies.
I think the Euler-Lagrange equation in the manifold case is given by "moving" the functional to a chart, i.e. multiplying the functional $\mathcal{L}$ with the volume element $\sqrt{|g|}$ for the metric tensor $g$ and its determinant $|g|$. Since $\sqrt{|g|}$ is a constant w.r.t. the functional derivative it can be moved out like this:
\begin{equation} \tag{1} \sqrt{|g|}\; \frac{\partial \mathcal{L}}{\partial f} - \sum_j \frac{\partial}{ \partial x_j} \; \sqrt{|g|} \;\frac{\partial \mathcal{L}}{\partial (\frac{\partial}{\partial x_j} f)} \; = \; 0 \end{equation}
(please correct me if I'm wrong).
Now I'm interested in minimizing w.r.t. natural boundary conditions. At https://physics.stackexchange.com/a/218894 I found the formula for the transversality condition
\begin{equation} \tag{2} n_{\mu}\frac{\partial {\cal L}}{\partial (\partial_{\mu}\phi^{\alpha})}=0 \end{equation}
(I assume this is meant as a sum in sense of Einstein notation, as I found it for the Euclidean case also in http://www.cmap.polytechnique.fr/~antonin/CaVa/Notes1.pdf, eq (2.7), page 11; it appears to be a draft though.)
It is stated for subsets of $\mathbb{R}^n$ so there might be missing some metric terms. OTOH the author calls it a "spacetime region" which suggests it holds not only for flat Euclidean metric but also for Einstein-Minkowski metric. So maybe there are really no metric terms involved here. Regarding a factor $\sqrt{|g|}$ before the functional I see that it can be omitted here because of the zero right hand side.
What I suspect is that the product with the normal might require to be carried out w.r.t. the metric scalar product as both $n_{\nu}$ and $\partial_{\nu}$ are written with lower indices. E.g. like
\begin{equation} \tag{3} n_{\nu} g^{\nu \mu} \frac{\partial {\cal L}}{\partial (\partial_{\mu}\phi^{\alpha})} = 0 \end{equation}
Are there no metric terms required because the functional derivative yields a covector while $n$ is a contravariant vector? With this post I request clarification and confirmation or correction of the formulas I posted.
Also a properly citable reference for these formulas would be appreciated. Books on variational calculus usually consider only the Euclidean case or no natural boundary conditions or directly move on to more advanced stuff. So far I found it surprisingly hard to find a reference for this case.