Past exam of the options II course of the University of Oxford / 2005 / G / 2 / i / Solution

Write $I[y]={\int }_a^{b}G(x,y,y^{\prime })dx$. For endpoint-zero variations $\eta$, the first variations areFor a normal constrained extremal, choose a variation ${\eta }_0$ with $\delta I({\eta }_0)\ne 0$. Given any $\eta$, use the implicit function theorem to choose a small correction in the ${\eta }_0$ direction so that $y+t\eta +s(t){\eta }_0$ preserves the constraint. Differentiating it and the stationarity condition at zero givesIntegrate the derivative-of-variation term by parts. Since the variations vanish at both endpoints, the boundary term is zero. The fundamental lemma of the calculus of variations then givesThe regularity condition on the constraint is what permits the coefficient of $F$ to be normalized to 1. In a degenerate case, the general multiplier statement uses ${\lambda }_{0}F-{\lambda }_{1}G$ with the two coefficients not both zero; the source's displayed form is the normal case.

For a string described as a function graph $r(x)=(x,y(x))$, its arc-length element is $|r^{\prime }(x)|dx=\sqrt{1+y^{\prime }(x{)}^2}dx$. Inextensibility therefore givesThis also forces $l\ge 2a$, with equality only for the straight function graph.