Natural boundary conditions for second-derivative functionals ($F_{y^{\prime }}-(F_{y^{\prime \prime }}{)}^{\prime }=0,F_{y^{\prime \prime }}=0$)

At an endpoint of a second-derivative variational problem, a free value $y$ gives the first displayed condition, while a free slope $y^{\prime }$ 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 $F=K(y^{\prime \prime }{)}^2/2+qy$, the two conditions become $y^{\prime \prime \prime }=0$ and $y^{\prime \prime }=0$, 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.