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I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu \frac{\partial L}{\partial (\partial_\mu \phi)}$$ Where $\phi$ is whatever field is an question and $\mu$ is (I believe) being summed from 0 to 3. Or, ist the correct equation $$\frac{\partial L}{\partial \phi}=D_\mu \frac{\partial L}{\partial (D_\mu \phi)}$$ Where $D$ is the covariant derivative of the theory. My intuition tells me its the second eqn but i just wanted to be sure, and I think I once saw someone say that the two were equivalent.

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I) Assuming that the variational problem for the action $S=\int \! d^nx~{\cal L}$ is well-posed (with appropriate boundary conditions), the field-theoretic Euler-Lagrange (EL) equations read in general

$$\tag{1} 0~\approx~\frac{\delta S}{\delta \phi^{\alpha}} ~=~\frac{\partial {\cal L}}{\partial \phi^{\alpha}} -\sum_{\mu} \frac{d}{dx^{\mu}} \frac{\partial {\cal L}}{\partial (\partial_{\mu}\phi^{\alpha})} + \sum_{\mu\leq \nu} \frac{d}{dx^{\mu}} \frac{d}{dx^{\nu}} \frac{\partial {\cal L}}{\partial (\partial_{\mu}\partial_{\nu}\phi^{\alpha})} - \ldots, $$

where the $\approx$ symbol means equality modulo eoms, and the ellipsis $\ldots$ denotes possible higher derivative terms. Note that the spacetime derivative

$$\tag{2} \frac{d}{dx^{\mu}}~=~ \frac{\partial }{\partial x^{\mu}} +\sum_{\alpha}(\partial_{\mu}\phi^{\alpha})\frac{\partial }{\partial \phi^{\alpha}} + \sum_{\alpha, \nu} (\partial_{\mu}\partial_{\nu}\phi^{\alpha})\frac{\partial }{\partial (\partial_{\nu}\phi^{\alpha})} + \ldots $$

is the total spacetime derivative rather than a partial spacetime derivative.

The version (1) of the EL equations is the basic formulation of the EL equations, which always works. Equation (1) holds even for non-covariant theories.

II) Now by imposing further conditions on the theory, such as,

  • it should be covariant in appropriate sense (e.g. gauge covariant, or general covariant under change of coordinates),

  • the appearances of spacetime derivatives in the Lagrangian should be minimally coupled via covariant derivatives,

  • etc,

it is often possible to derive versions of the EL equations where spacetime derivatives $\partial_{\mu}$ and $\frac{d}{dx^{\mu}}$ have been replaced with covariant derivative counterparts of appropriate type (e.g. gauge-type or gravity-type covariant derivatives).

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