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: $$\sum_{i=1}^{4}\frac{\partial}{\partial x_{i}}\bigg{(}\frac{\partial \mathcal{L}}{\partial (\partial_{x_{i}}\phi)}\bigg{)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0 $$

Action Principle and PDEs(Princeton Univ. Press). But I don't have my copy with me to give you a precise page reference. $\endgroup$ – Willie Wong Jan 29 at 21:33canbe made rigorous (see e.g. mathoverflow.net/a/349234/11211, mathoverflow.net/q/273254/11211 and physics.stackexchange.com/a/256496/16767). The problem is just the terminology employed in (some of) the physical literature - one is in fact looking forcritical pointsof afamilyof action functionals over each compact region of space-time. Once properly defined, these arebona fidesmooth functionals on the space of smooth field configurations, which on its turn can be endowed with a proper infinite-dimensional smooth manifold structure. $\endgroup$ – Pedro Lauridsen Ribeiro Feb 2 at 2:401more comment