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Classical field theory and the Euler-Lagrange equations

In this post I'll try to briefly give a more-precise-than-you-usually-find explanation of the Euler-Lagrange equations for classical fields. This is a point where textbooks tend to be sloppy so hopefully this post can clear things up somewhat, even though I will ignore the analytical details of convergence and smoothness. In a follow-up post I plan to explain Noether's theorem for fields. Let's begin with the simplest example: a real scalar field. That is simply a function $\phi : \mathbb{R}^{n+1} \to \mathbb{R}$ (assumed to be sufficiently smooth and to decay sufficiently fast at infinity). Here $\mathbb{R}^{n+1}$ stands for one dimension of time and $n$ dimensions of space. We write the coordinates on $\mathbb{R}^{n+1}$ as $x^\mu$, where $\mu$ ranges from $0$ to $n$, with $\mu = 0$ being the time dimension. One obtains the equations of motion for $\phi$ from a Lagrangian density $\mathcal{L}$. In this case, the Lagrangian density is a function [1] $\mathcal{L} : \mat