I'm looking for **rigorous** discussions on the derivation of the Euler-Lagrange equation for **field** as it is usually discussed in classical field theory books. More precisely, if the action is given by:


$$S(\phi) = \int \mathcal{L}(\phi, \partial\_{x\_{i}}\phi) d^{4}\vec{x}$$
where $\vec{x} = (x\_{1},x\_{2},x\_{3},x\_{4}=t)\in \mathbb{R}^{4}$ and $\partial\_{x\_{i}}$ denotes, generically, any of its partial derivatives, then I'm looking for a rigorous derivation of the Euler-Lagrange equation:
$$\sum\_{i=1}^{4}\frac{\partial}{\partial x\_{i}}\bigg{(}\frac{\partial \mathcal{L}}{\partial (\partial\_{x\_{i}}\phi)}\bigg{)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0 $$



Rigorous Euler-Lagrange equations for fields
See page 16 and following of [Coordinate-free derivation of the Euler–Lagrange equations and identification of global solutions via local behavior](https://mast.queensu.ca/~andrew/papers/pdf/2005a.pdf) by Elsa Hansen (2005).



> 
> Results concerning $C^2$-minimizing curves on manifolds are presented.
> A coordinate- free derivation of the Euler–Lagrange equation is
> presented. Using a variational approach, two vector fields are defined
> along the minimizing curve; the tangent to the curve $\dot{γ}$, and
> the infinitesimal variation $\delta\sigma$. The derivation presented
> involves complete lifts of arbitrary extensions of these vector fields
> and it is shown that the derivation is independent of the particular
> choice of extensions. Special care is also taken to ensure that the
> derivation does not require any additional differentiability
> constraints, other than $\gamma$ being of class $C^2$.
> 
> 
> 


