The integral

$$I=\int_{a}^{b} F\left(y(x), y^{\prime}(x)\right) d x,$$

where $F$ is some functional, is defined for the class of functions $y(x)$ for which $y(a)=y_{0}$, with the value $y(b)$ at the upper endpoint unconstrained. Suppose that $y(x)$ extremises the integral among the functions in this class. By considering perturbed paths of the form $y(x)+\epsilon \eta(x)$, with $\epsilon \ll 1$, show that

$$\frac{d}{d x}\left(\frac{\partial F}{\partial y^{\prime}}\right)-\frac{\partial F}{\partial y}=0$$

and that

$$\left.\frac{\partial F}{\partial y^{\prime}}\right|_{x=b}=0 .$$

Show further that

$$F-y^{\prime} \frac{\partial F}{\partial y^{\prime}}=k$$

for some constant $k$.

A bead slides along a frictionless wire under gravity. The wire lies in a vertical plane with coordinates $(x, y)$ and connects the point $A$ with coordinates $(0,0)$ to the point $B$ with coordinates $\left(x_{0}, y\left(x_{0}\right)\right)$, where $x_{0}$ is given and $y\left(x_{0}\right)$ can take any value less than zero. The bead is released from rest at $A$ and slides to $B$ in a time $T$. For a prescribed $x_{0}$ find both the shape of the wire, and the value of $y\left(x_{0}\right)$, for which $T$ is as small as possible.