= Solution
An Euler strut is a slender elastic column under axial compression. Let its <bending stiffness> about the weaker section axis be $EI_A$ and its <length> be $L$. With small lateral deflection $y(x)$ and a dead compressive load $P$, the quadratic total <potential energy> is
$$
\Pi[y]=\frac12\int_0^L EI_A(y'')^2\,dx
-\frac P2\int_0^L(y')^2\,dx.
$$
The first term is bending <energy>; the negative term is <work> released by the load as bending shortens the end-to-end distance. Stationarity gives $EI_Ay''''+Py''=0$. For pinned ends, $y=y''=0$ at $0,L$, and $y\propto\sin(n\pi x/L)$ requires $P_n=n^2\pi^2EI_A/L^2$. The first instability is <Euler's critical load>,
$$
\boxed{P_{\rm cr}=\frac{\pi^2EI_A}{L^2}.}
$$
Below this load, the straight state has positive quadratic <energy> against these perturbations. At the threshold, the lowest bending mode has zero quadratic restoring <energy>; above it the straight state is unstable. For a rectangular section, $I_A=bh^3/12$ about the appropriate bending axis, so the smaller <second moment of area> controls the first buckling direction.
Endpoint <constraints> matter. Write $P_{\rm cr}=\pi^2EI_A/(KL)^2$: $K=1$ for pinned-pinned, $2$ for fixed-free and $1/2$ for fixed-fixed conditions. This elastic instability differs from compressive material failure. Imperfections cause bending before the ideal threshold, and a short column may yield before <Euler buckling> occurs.
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