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Let $S_W$ be a Wilson action of $1\times 1$ plaquettes for a gauge group $G$: \begin{equation*} S_W = \beta a^4 \sum_P \left( 1-\frac{1}{N_G} \text{Re Tr}(U_P) \right), \end{equation*} where $\beta$ is a positive constant, $a$ is the lattice parameter, $N_G$ is the dimension of the matrices of the fundamental representation of $G$, and the $U_P$ are defined as: \begin{equation} U_P \equiv \mathcal{P} e^{i \oint_{\partial P} A}. \end{equation} For a path ordering operator $\mathcal{P}$ and a gauge $\mathfrak{g}$-valued 1-form $A$. My question is: What are the equations of motion of such action? In particular I don't know if I can use the Euler-Lagrange equations since we are on a discrete set of points.

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