= Euler-Lagrange field equation with second derivatives
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{title2=$\mathcal L_\phi-\partial_\mu\mathcal L_{\phi_{,\mu}}+\partial_\mu\partial_\nu\mathcal L_{\phi_{,\mu\nu}}=0$}
For an <action> $S=\int\mathcal L(\phi,\partial\phi,\partial^2\phi)d^{d+1}x$, vary the field with compact support, or fix both $\delta\phi$ and its first derivatives on the boundary. One <integration by parts> on the first-derivative term and two on the second-derivative term give the displayed <Euler-Lagrange field equation>. With ordered repeated indices, define the second-derivative coefficient symmetrically so that $\delta\mathcal L=\mathcal L_\phi\delta\phi+\mathcal L_{\phi_{,\mu}}\partial_\mu\delta\phi+\mathcal L_{\phi_{,\mu\nu}}\partial_\mu\partial_\nu\delta\phi$. This convention avoids an unintended factor of two in mixed derivative terms. Higher spatial derivatives alone do not require higher time derivatives.
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