Euler-Lagrange field equation with second derivatives (${\mathcal{L}}_{\varphi }-{\partial }_{\mu }{\mathcal{L}}_{{\varphi }_{,\mu }}+{\partial }_{\mu }{\partial }_{\nu }{\mathcal{L}}_{{\varphi }_{,\mu \nu }}=0$)

For an action $S=\int \mathcal{L}(\varphi ,\partial \varphi ,{\partial }^2\varphi )d^{d+1}x$, vary the field with compact support, or fix both $\delta \varphi$ 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 $\delta \mathcal{L}={\mathcal{L}}_{\varphi }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu }}{\partial }_{\mu }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu \nu }}{\partial }_{\mu }{\partial }_{\nu }\delta \varphi$. This convention avoids an unintended factor of two in mixed derivative terms. Higher spatial derivatives alone do not require higher time derivatives.