Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 1 / b / Solution

In the dark, energy minimization gives $h_{xx}=-H_0$. Translation and rotation are zero-energy freedoms; one representative is $h_{-}(x)=-H_0{x}^2/2$. After illumination, choose the equivalent light-adapted equilibriumThen $u=h-h_{+}$ has homogeneous free-end conditions $u_{xx}=u_{xxx}=0$, and its initial valueis orthogonal to the two rigid zero modes $1$ and $x$.

Let $q_n>0$ solve the free--free biharmonic eigenvalue equationand chooseThese are orthogonal eigenfunctions of the biharmonic operator with ${\varphi }_n^{\prime \prime }={\varphi }_n^{\prime \prime \prime }=0$ at both ends. The shape iswhereThe omitted zero modes would only translate or rotate the whole filament.