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

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  • $\begingroup$ What is unsatisfactory with the usual derivation that makes them non-rigorous? (Specifically, are you worried about the use of the compactly support perturbations? Are you worried about the fact that $S(\phi)$ is, for most field theories, necessarily infinite?) $\endgroup$ – Willie Wong Jan 29 at 2:09
  • $\begingroup$ @WillieWong this is one reason. But also, all derivations I know come from physics books and thus there are some calculuations which seem unclear or poorly justified to me. $\endgroup$ – MathMath Jan 29 at 13:00
  • $\begingroup$ Most of my books are in my office; hopefully Igor Khavkine sees your question since I know for sure he has an answer to it. The correct formulation is also described in his paper arxiv.org/abs/1210.0802 but I don't know how much you would like the Jet language. I am also pretty sure that this is explained in Christodoulou's 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 21:33
  • $\begingroup$ I don't think there is a rigorous derivation. For one thing, it is possible to construct systems in which the physically-correct time evolution does not correspond to a minimization of the action. It could be a maximum of the action or—more crucially—a saddle point of the action. But what does it mean, precisely speaking, to be a saddle point of the action? Well, the fields obey the Euler-Lagrange equations.... $\endgroup$ – Buzz Feb 1 at 22:01
  • $\begingroup$ @Buzz the derivation can be 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 for critical points of a family of action functionals over each compact region of space-time. Once properly defined, these are bona fide smooth 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 2:40
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See page 16 and following of Coordinate-free derivation of the Euler–Lagrange equations and identification of global solutions via local behavior 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 $C^2$.

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  • $\begingroup$ The reference concerns minimizing curves rather than submanifolds, and thus concerns mechanics rather than field theory. But the method of the proof is similar in both cases. $\endgroup$ – Mikhail Skopenkov Feb 5 at 22:11

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