= Zero-energy filament mode
{title2=$K\psi=0$}
A zero-energy mode lies in the <kernel> of the quadratic <elastic filament> operator. For pure bending, zero curvature makes the displacement affine. Free-free endpoints therefore leave translation and tilt modes, while slope-constrained, force-free endpoints leave only translation. Both clamped-clamped and hinged-hinged endpoints remove these modes. An unrestricted modal amplitude has constant <Boltzmann distribution> weight, so no normalizable <canonical ensemble> or finite displacement <variance> exists. Fixing the rigid degrees of freedom allows the <equipartition theorem> on the positive complement.
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