= Natural boundary condition on the slope
{title2=$F_{y''}=0$}
= Natural boundary conditions on the slope
{synonym}
For a <functional> $J[y]=\int F(x,y,y',y'')\,dx$, two applications of <integration by parts> leave the boundary term
$$
\left[\left(F_{y'}-\frac d{dx}F_{y''}\right)\delta y+F_{y''}\delta y'\right]_a^b.
$$
If the endpoint displacement is fixed but its slope is free, $\delta y=0$ while $\delta y'$ is arbitrary. The <first variation> therefore forces $F_{y''}=0$ at that endpoint. For the beam energy density $F=\beta(y'')^2/2-\alpha(y')^2/2$ with $\beta>0$, this gives $y''=0$, the zero-<bending moment> condition. A clamped endpoint instead prescribes the slope and has $\delta y'=0$.
Back to article page
