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 variabled $\phi$ and $\psi$ is given by $$ S(\phi, \psi) = \int_M h(\phi, \psi)\, \mathrm{d}V_M $$ Now, I would like to find a field equations associated with $S$, and it's equivalent to find the local expression of the 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.
EDIT In fact, the question is that how can we deal with the field equations on smooth manifold if the field is considered as the section of some hermitian line bundle.
I know that there is a similar question in MO, but I'm still wondering how to get such field equations from the globally defined action functional on the bundle, and solve them. Is it possible without any specific local expression for general case?
or any references about this subject is also appreciated. I think I'm too ignorant of the subject.
