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The functional derivative is $\delta \mathcal{E}/\delta h=A{h}^{(4)}$, so the overdamped filament bending equation is $\zeta h_t=-A{h}^{(4)}$ with the free-end conditions from (a). Expand the initial shape in the complete orthonormal basis:The overdamped relaxation of a free filament isEach bending mode relaxes on time ${\tau }_n=\zeta /(A{k}_n^4)$; the two zero modes remain constant. Consequently the long-time shape is the affine projection of $H$,The total displacement and its first moment are conserved: integrate $h_t$ and $(x-L/2)h_t$, using $h^{\prime \prime }=h^{(3)}=0$ at the ends. This also checks why the final affine part cannot generally be set to zero. The series defines the $L^2$ solution for arbitrary square-integrable initial data; a classical solution at time zero additionally requires compatible boundary values.