Self-adjoint endpoint conditions for filament bending

For the elastic filament operator $K=A{\partial }_x^4$, two integration by parts operations give boundary form $A[\overline{u}{v}^{\prime \prime \prime }-\overline{u^{\prime }}{v}^{\prime \prime }+\overline{u^{\prime \prime }}{v}^{\prime }-\overline{u^{\prime \prime \prime }}v]$. Four elementary homogeneous choices at each endpoint are $h=h^{\prime }=0$ (clamped), $h=h^{\prime \prime }=0$ (hinged), $h^{\prime \prime }=h^{\prime \prime \prime }=0$ (free), and $h^{\prime }=h^{\prime \prime \prime }=0$ (slope-constrained and transverse-force-free). Each defines a self-adjoint operator on a finite interval when imposed at both ends. More general real Robin boundary conditions also give self-adjointness; the four choices are not an exhaustive classification of all boundary subspaces.

Variation of the small-slope bending energy of an elastic filament gives endpoint work $A[h^{(2)}\delta h^{(1)}-h^{(3)}\delta h]$. Arbitrary endpoint displacement and slope variations force the bending moment and transverse force to vanish. These are $h^{(2)}=h^{(3)}=0$. They define a self-adjoint nonnegative fourth-order bending operator on a finite interval; its kernel contains rigid translation and tilt.

Nonzero bending modes have the sine, cosine, hyperbolic sine and hyperbolic cosine form. Free conditions at the first end identify the cosine and hyperbolic cosine coefficients, and likewise the sine coefficients. The remaining boundary determinant is $2(1-\cos q\cosh q)$. Its first positive roots are about $4.73004,7.85320,10.99561$. The complete basis additionally includes two zero-energy filament modes, constant translation and affine tilt; these are absent from the finite-coefficient formula at $k=0$.

Writing the characteristic equation with reciprocal hyperbolic cosine avoids exponentially large products. For the positive roots indexed from one, $q_n=(n+1/2)\pi +2(-1{)}^{n+1}{e}^{-(n+1/2)\pi }+O(e^{-2(n+1/2)\pi })$. The apparent root at zero represents the separate affine kernel, not another positive bending mode. Evaluating hyperbolic pieces of high modes with decaying exponential forms also avoids cancellation.