= Endpoint position compatibility for the Euler elastica
{title2=$\Delta y=\int_0^L\sin\theta\,ds$}
Specifying endpoint tangent angles of an <inextensible rod> does not by itself specify its endpoint positions. A planar <Euler elastica> satisfies $x'=\cos\theta$, $y'=\sin\theta$, so equal endpoint heights require the displayed <integral> to vanish. For constant compressive resultant $(-P,0)$, the angular equation is $EI\theta''+P\sin\theta=0$, giving $P\Delta y=-EI[\theta'(L)-\theta'(0)]$. Thus a nontrivial branch containing an odd number of pendulum half-periods cannot have equal endpoint heights and the same zero endpoint angles with zero transverse resultant. At small amplitude the tangent-only <eigenfunctions> are $\sin(n\pi s/L)$, with $P_n=EI(n\pi/L)^2$; equal endpoint heights retain even $n$ and give the first fixed-clamp load $4\pi^2EI/L^2$. Other clamp modes can require a nonzero transverse reaction. Axial shortening is also needed when the reference endpoint separation equals the rod length.
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