Restricted variation and the full field equations

Let ${\mathcal{C}}_r,\mathcal{C},S,I,\varphi ,\psi =I(\varphi )$ and the topological direct sum $T_{\psi }\mathcal{C}=V_{\mathrm{r}\mathrm{e}\mathrm{t}}(\varphi )\oplus V_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}(\varphi )$ be as in the supplied definition. Set $S_r=S\circ I$. Call $\psi$ stationary for $S$ when $DS{\mathrm{\mid }}_{\psi }=0$ on all admissible full variations, and call $\varphi$ stationary for $S_r$ when $D{S}_r{\mathrm{\mid }}_{\varphi }=0$. Then $D{S}_r{\mathrm{\mid }}_{\varphi }=(D{I}_{\varphi }{)}^{\ast }(DS{\mathrm{\mid }}_{\psi }),$ so $D{S}_r{\mathrm{\mid }}_{\varphi }=0$ if and only if $DS{\mathrm{\mid }}_{\psi }$ annihilates $V_{\mathrm{r}\mathrm{e}\mathrm{t}}(\varphi )$. The configuration $\psi$ is stationary for $S$ if and only if the restrictions of $DS{\mathrm{\mid }}_{\psi }$ to both $V_{\mathrm{r}\mathrm{e}\mathrm{t}}(\varphi )$ and $V_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}(\varphi )$ vanish. Therefore reduced stationarity implies full stationarity exactly when the omitted restriction also vanishes. No local Euler–Lagrange differential operator is asserted here; for an action whose boundary terms vanish, these are its variational stationarity conditions.

Retained and omitted variation spaces