 Regularity of a normalized quadratic variational constraint

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 $2\int w{y}^2=2$, so the constraint is always regular at an admissible function. With free endpoint values this justifies the Euler-Lagrange equation for $F-\lambda w{y}^2$ and its natural endpoint conditions. For $F=p{y}^{\prime 2}-q{y}^2$, multiplication of the differential equation by $y$ gives $\lambda =J[y]$.