Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 355 / 3 / a / Solution

For an inextensible planar filament, ${\mathbf{r}}_s=(\cos \theta ,\sin \theta )$. The bending energy and compressive-force potential areThe straight state has $E=FL$. Free pivoting means zero end moment,The endpoints sharing the $x$ axis impose ${\int }_0^L\sin \theta ds=0$; near the straight state this removes the constant angular mode. The Neumann eigenfunctions therefore give