= Euler-Lagrange field equation
The <Euler-Lagrange field equation> implements the <principle of stationary action> for a <Lagrangian density> depending on <scalar fields> and their first <derivatives>. For a single <scalar field> $\Phi$,
$$
\frac{\partial\mathcal L}{\partial\Phi}
-\partial_\mu\frac{\partial\mathcal L}{\partial(\partial_\mu\Phi)}=0.
$$
Vary $S=\int d^4x\,\mathcal L$. <Integration by parts> transfers the <derivative> from $\delta\Phi$ to $\partial\mathcal L/\partial(\partial_\mu\Phi)$. The <boundary term> vanishes for variations supported away from the boundary; arbitrary interior variations then force the displayed coefficient to vanish. For $\mathcal L=\tfrac12\partial_\mu\Phi\,\partial^\mu\Phi-\tfrac12m^2\Phi^2$, the result is the <Klein-Gordon equation> $(\partial_\mu\partial^\mu+m^2)\Phi=0$, using <natural units> and <Minkowski metric> signature $(+---)$. <Quantum Field Theory by David Tong>[Sections 1.1–1.1.1, equations 1.6–1.15]{ref}
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