= Critical-load minimality of the angular Euler strut
{title2=$\lambda=1:\quad U[\theta]>U[0]\text{ for }\theta\ne0,\quad\delta^2U[0]\text{ is degenerate}$}
For the angular endpoint conditions $\theta(0)=\theta(1)=0$, take $U=(EI/L)\int_0^1[\theta'^2/2-\pi^2\lambda(1-\cos\theta)]\,d\xi$. The <Poincaré inequality> and $1-\cos t\leq t^2/2$ give $U\geq0$ for $0\leq\lambda\leq1$. At $\lambda=1$ a nonzero continuous angle still has strictly positive energy because $1-\cos t<t^2/2$ for $t\ne0$. Thus the straight state remains the unique global minimum at the critical parameter even though the quadratic variation vanishes on the first sine mode.
Along that mode the energy is $\pi^2EI A^4/(64L)+O(A^6)$ at criticality. For $\lambda>1$ a sufficiently small first-mode variation lowers energy, so the straight state loses local minimality. Near the first threshold, expansion of the Euler strut equation gives $A_0(A_0^2-8\lambda_1)=0$ when $\lambda=1+\epsilon\lambda_1$ and $\theta\sim\sqrt\epsilon A_0\sin\pi\xi$. The resulting <supercritical pitchfork bifurcation> is stabilized by the same positive quartic term. A zero second variation is therefore an inconclusive quadratic test, not proof of failure of minimization.
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