Suppose $M$ be any smooth manifold, and $E$ be a hermitian line bundle over $M$ with hermitian metric $h$. The action functional $S(\phi, \psi)$ for two section-variable $\phi, \psi$ are given by
$$
S(\phi, \psi) = \int_M h(\phi, \psi) \mathrm{d}V_M
$$
Now, I would like to find a field equations associated to $S$, and it's equivalent to find the local expression of Euler-Lagrange equation on each charts.

Let $U$ be a chart of $M$, and take a trivialization $\Phi$ of $E|U \cong U \times \mathbb{C}$ such that $\Phi^* h$ is just a trivial complex sesquilinear form. 
Suppose $\varphi^\ast$ and $\varphi$ be a local representation of $\psi$ and $\phi$.
Then, the local Euler-Lagrange equation associated to $S$ is equivalent to
$$
\varphi^* = 0\\
\varphi = 0
$$ since the local Lagrangian is given by $\mathcal{L}|\Phi = \varphi^* \varphi$.

The action functional is too simple here and so the equations also become too simple, but in fact, this is just a toy model for checking whether my approach is correct or not.

Any comments, improvements and advices are appreciated.