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

In the Monge representation, $ds=dx+O(h_x^2)$ and the signed curvature is $\kappa =h_{xx}+O(h_x^2{h}_{xx})$. To quadratic order,Two integrations by parts giveThus a filament with free ends has zero bending moment and shear force,and the local Stokesian dynamics of an elastic filament isWith $X=x/L$, $H=h/L$, and $\tau =At/(\zeta L^4)$, the dimensionless problem isat $X=0,1$.