= Bessel threshold for self-buckling of a vertical rod
{c}
{title2=$\lambda gL_c^3/(EI)=9j_{-1/3,1}^2/4$}
For a uniform <elastic filament> clamped below and free above, its own weight gives <filament tension> $T(z)=-\lambda g(L-z)$. With <filament bending modulus> $B=EI$, the static slope $u$ satisfies $Bu''+\lambda g(L-z)u=0$, $u(0)=0$, and $u'(L)=0$. Setting $\eta=(2/3)\sqrt{\lambda g/B}(L-z)^{3/2}$ gives $u=\eta^{1/3}J_{-1/3}(\eta)$ up to amplitude; the $J_{1/3}$ branch violates the zero-moment condition. The first positive zero $j_{-1/3,1}$ therefore yields $\lambda gL_c^3/B\simeq7.83735$. This is a <self-buckling of a vertical rod> threshold, not a constant-end-load Euler threshold.
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