 Solution

For the elastic filament, apply a variation $\eta (x)$. Two integration by parts operations giveAt a free endpoint both $\eta$ and ${\eta }^{\prime }$ are arbitrary. The free-end bending boundary conditions and the unloaded Euler-Lagrange equation are thereforeHere $A{h}^{\prime \prime }$ is the bending moment and $-A{h}^{(3)}$ the transverse shear-force convention. The equilibrium shapes are affine, reflecting free translation and tilt.

On $L^2(0,L)$ define $K=A{\partial }_x^4$ on $H^4$ functions obeying those endpoints. The self-adjoint endpoint conditions for filament bending annul the boundary formThis proves symmetry. To establish actual self-adjointness, the adjoint-domain boundary form must vanish for every free-end $u$. The endpoint values $u,u^{\prime }$ can be varied independently, forcing $v^{\prime \prime }=v^{(3)}=0$ at both ends. The regular fourth-order expression then gives the same $H^4$ domain for the adjoint. Thus the domains coincide and $K$ is self-adjoint. Also $⟨h,Kh⟩=A\int |h^{\prime \prime }{|}^{2}dx\ge 0$.