Euler-Lagrange field equation

For a field $\Phi$,

An off shell field configuration is not required to satisfy the Euler-Lagrange field equation. An external four-momentum is off shell if it is not constrained to its physical mass shell. On shell identities can therefore fail for general untransformed correlation functions, even when the associated S-matrix is unchanged by a field redefinition.

Adding $({\varphi }_1{\varphi }_{2,t}-{\varphi }_2{\varphi }_{1,t})/(2\tau )$ to the cross-coupled scalar Lagrangian density produces independent equations ${\varphi }_{1,tt}-c^2{\varphi }_{1,xx}+{\varphi }_{1,t}/\tau =0$ and ${\varphi }_{2,tt}-c^2{\varphi }_{2,xx}-{\varphi }_{2,t}/\tau =0$. For the convention $e^{i(kx-\omega t)}$, their frequencies are respectively $-i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$ and $+i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$. Above the threshold they oscillate with decaying or growing envelopes; below it they have nonoscillatory real temporal exponents. At equality the repeated temporal root supplies $(A+Bt)e^{\mp t/(2\tau )}$. At $k=0$ each equation also has a constant solution. The full system is generically unstable because of the amplified second field, even though it arises from an autonomous variational system with conserved, indefinite total energy.

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.

A field configuration or identity is on shell when the equations of motion hold.