Write down the Euler-Lagrange equation for extrema of the functional

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

Show that a first integral of this equation is given by

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

A road is built between two points $A$ and $B$ in the plane $z=0$ whose polar coordinates are $r=a, \theta=0$ and $r=a, \theta=\pi / 2$ respectively. Owing to congestion, the traffic speed at points along the road is $k r^{2}$ with $k$ a positive constant. If the equation describing the road is $r=r(\theta)$, obtain an integral expression for the total travel time $T$ from $A$ to $B$.

[Arc length in polar coordinates is given by $d s^{2}=d r^{2}+r^{2} d \theta^{2}$.]

Calculate $T$ for the circular road $r=a$.

Find the equation for the road that minimises $T$ and determine this minimum value.