= Spatially varying tension in filament bending
{title2=$K_\sigma=A\partial_x^4-\partial_x(\sigma\partial_x)$}
Adding $\frac12\int\sigma(x)(h')^2dx$ to the quadratic bending energy of an <elastic filament> contributes $-(\sigma h')'$ to its <variational derivative>. The endpoint term is $Ah''\eta'+(\sigma h'-Ah''')\eta$. If the real <filament tension> vanishes at both ends, the usual <self-adjoint endpoint conditions for filament bending> remain valid. A complete <eigenfunction expansion> diagonalizes the energy even when the <eigenfunctions> have no explicit formula. Thermal <equipartition theorem> arguments require positive eigenvalues after fixing any <kernel>; strong compression can violate this requirement.
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