= Compressed Euler-Bernoulli beam and its endpoint-dependent buckling threshold
{title2=$\rho\phi_{tt}+\alpha\phi_{xx}+\beta\phi_{xxxx}=0$}
Positive compression $\alpha$ contributes negative quadratic <potential energy> $-\alpha\phi_x^2/2$, while positive <bending stiffness> $\beta$ contributes $\beta\phi_{xx}^2/2$. A <plane wave> has $\omega^2=(\beta k^4-\alpha k^2)/\rho$, so sufficiently long waves grow exponentially. For displacement-fixed, moment-free endpoints at $0,l$, the <natural boundary conditions for a free endpoint> on the slope give $\phi_{xx}=0$ as well as $\phi=0$. The modes are $\sin(n\pi x/l)$, and the first <Euler buckling of an elastic filament> threshold is $\alpha_c=\beta\pi^2/l^2$. For truly clamped ends, $\phi=\phi_x=0$, the threshold is instead $4\pi^2\beta/l^2$, with first static mode $1-\cos(2\pi x/l)$. More generally,
$$
\alpha_c=\beta\inf_{\phi\ne0}\frac{\int_0^l\phi_{xx}^2dx}{\int_0^l\phi_x^2dx},
$$
with the infimum taken over the specified endpoint class. At threshold there is a static zero mode and a marginal linear-in-time partner in the linear dynamics; above it the straight configuration is unstable.
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