= Solution
Take $A>0$ and use primes for spatial derivatives. Two <integration by parts> operations in the first variation of the <elastic filament>'s bending energy give
$$
\delta\mathcal E=A\int_0^L h''''\eta\,dx+A[h''\eta'-h'''\eta]_0^L.
$$
Thus the <Euler-Lagrange equation> is $Ah''''=0$, and the fluctuation <differential operator> is $K=A\,d^4/dx^4$. In the <L2 space> <inner product>, its boundary form is
$$
\langle u,Kv\rangle-\langle Ku,v\rangle=A[\overline u v'''-\overline{u'}v''+\overline{u''}v'-\overline{u'''}v]_0^L.
$$
The conjugate endpoint trace pairs are $(h,h''')$ and $(h',h'')$. Requiring one member of each pair to vanish gives the four standard <self-adjoint endpoint conditions for filament bending>, applied at both ends:
* \b[Free-free:] $h''=h'''=0$. Both the bending <torque> and the transverse endpoint <force> vanish; position and slope can vary.
* \b[Clamped-clamped:] $h=h'=0$. Position and slope are fixed, with reaction <forces> and <torques> permitted. These are <clamped boundary conditions>.
* \b[Hinged-hinged:] $h=h''=0$. Position is fixed, but the endpoint rotates without bending <torque>.
* \b[Torqued-torqued:] $h'=h'''=0$. Slope is fixed by an endpoint <torque>, while translation is free and transverse <force> vanishes. The <torque> is a reaction, not an additional condition setting $h''$ to zero.
Each pair annihilates the boundary form for all $u,v$ in the domain. Conversely, the remaining two endpoint traces can be chosen freely: requiring the boundary form to vanish against every such $u$ forces an adjoint-domain function $v$ to satisfy the same two conditions. This proves <self-adjointness>, rather than just formal symmetry, on the corresponding fourth-order <Sobolev space> domain.
The count four concerns these elementary homogeneous choices. \b[Identical-end <boundary conditions> do not restrict all <self-adjoint operators> to these four possibilities.] For example, $h=0$, $h''=b h'$ at both ends, with any fixed real $b\ne0$, also annihilates the boundary form: the remaining expression is $-\overline{u'}b v'+b\overline{u'}v'=0$. This <Robin boundary condition> supplies a continuous family beyond the four listed pairs.
Back to article page
