Hypothesis and convention boundaries for the variational prerequisites

Two exact boundary checks. (i) Stationarity is first order. On ${\mathbb{R}}^2$ the polynomial $S(x,y)=x^2-y^2$ has differential $DS(0,0):{\mathbb{R}}^2\to \mathbb{R}$ equal to the zero map at $(0,0)$ in the total-derivative sense of The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, while $S(\epsilon ,0)={\epsilon }^2>0$ and $S(0,\epsilon )=-{\epsilon }^2<0$ for every $\epsilon \mathrm{\ne }0$; a point where the first variation vanishes need not be a minimum. Consequently the equivalence of Compact variations of the quadratic field functional: first variation and the coefficient equation between stationarity and the coefficient equation, and the conversion by The fundamental lemma of the calculus of variations for continuous coefficients of a vanishing first variation into pointwise equations, are first-order conditions: they carry no minimality and no existence content, and the example involves no adopted physical action. (ii) Compact support of the variation is essential. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for this Lebesgue-functional branch, exactly as in the compact-field supplier; the polynomial branch (i) remains choice-free. Take $U=(-1,1{)}^4$, $\kappa =1$, $J=0$ and the smooth field $A$ with $A_0(x)=(x^1{)}^2$ and $A_1=A_2=A_3=0$, in the conventions of The Minkowski affine four-space: the form $\eta$, causal cones, frames and Lorentz maps. Then $F_{01}=-2x^1$, $F_{10}=2x^1$, $F^{01}={\eta }^{00}{\eta }^{11}{F}_{01}=2x^1$ and $F^{10}=-2x^1$, with every remaining component vanishing, so ${\mathrm{\partial }}_{\mu }{F}^{\mu 0}={\mathrm{\partial }}_1{F}^{10}=-2\mathrm{\ne }0$ and the coefficient equation ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-\kappa J^{\nu }$ fails at $\nu =0$. Nevertheless the variation $h$ with $h_0\equiv 1$ and $h_1=h_2=h_3=0$ is smooth but not compactly supported in $U$; along it $A_0+s{h}_0=(x^1{)}^2+s$, so $F_{01}$ and the integrand $L=2(x^1{)}^2$ are independent of $s$ and $S[A+sh]={\int }_{U}2(x^1{)}^{2}d\lambda$ is constant in $s$, with first variation $0$ although the field equation fails. For the complementary stationarity-boundary check take instead $A_0=x^1$, the other covariant components zero, $J=0$, and $h_0=x^1$, $h_i=0$ (so $A^0=-x^1$, $h^0=-x^1$). Here $F_{01}=-1$ is constant, so the coefficient equations all hold, but this smooth noncompact direction gives $S[A+sh]=8(1+s{)}^2$ and derivative $16\mathrm{\ne }0$ at zero. Thus extending the compact stationarity condition to all smooth noncompact directions would invalidate the converse. These two witnesses show that the stated compact-support hypothesis cannot simply be removed: its coefficient conversion has no boundary term precisely in its declared compact class. Other variation classes require their own boundary hypotheses.