= Overdamped relaxation of a free filament
{title2=$h=\sum_{j\in\{\mathrm{tr},\mathrm{tilt}\}}c_jW_j+\sum_{n\geq1}c_nW_ne^{-Ak_n^4t/\zeta}$}
The <overdamped filament bending equation> diagonalizes in the complete <free-free bending spectrum>. Positive modes decay at rates $Ak_n^4/\zeta$, while translation and tilt remain fixed. The long-time shape is the affine least-squares projection of the initial height. Free-end boundary values conserve the total height integral and its first spatial moment. For square-integrable initial data the expansion gives the smoothing semigroup solution.
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