= Regularity of a normalized quadratic variational constraint
{title2=$K[y]=\int_a^bwy^2=1\ \Longrightarrow\ DK[y](y)=2$}
An integral constraint need not admit a normal <Lagrange multiplier> unless its <first variation> is nonzero. For the quadratic normalization in the title, the variation in direction $y$ has derivative $2\int wy^2=2$, so the constraint is always regular at an admissible function. With free endpoint values this justifies the <Euler-Lagrange equation> for $F-\lambda wy^2$ and its natural endpoint conditions. For $F=py'^2-qy^2$, multiplication of the differential equation by $y$ gives $\lambda=J[y]$.
Back to article page
