= Natural boundary conditions for second-derivative functionals
{title2=$F_{y'}-(F_{y''})'=0,\quad F_{y''}=0$}
At an endpoint of a second-derivative variational problem, a free value $y$ gives the first displayed condition, while a free slope $y'$ gives the second. Each condition is required only when its corresponding endpoint variation is free; a prescribed value or slope instead forces that variation to vanish. For $F=K(y'')^2/2+qy$, the two conditions become $y'''=0$ and $y''=0$, respectively. A clamped beam has both value and slope fixed, a pinned beam fixes value but leaves slope free, and a free end leaves both free.
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