Discontinuous Solutions. One-Sided Variation

Find the discontinuous solutions with a corner point for the problem of the minimum of the integral

Among the lines going from the point $A(0,0)$ to the point $B(2,0)$, find the line with a corner point along which the integral

attains its minimum value.

Find the discontinuous solutions with a corner point for the problem of the minimum of the integral

For the problem of the extremum of the integral

find the continuous solutions, as well as the solutions with a corner point.

Among all cylindrical surfaces with generators parallel to $OZ$, find the one on which the plane $z=0$, the cone $z = \sqrt{x^{2}+y^{2}}\operatorname {tg}a$, and the two generators passing through the points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$ cut off the smallest area.

In the plane $XOY$, find the curve joining two points $M_{1}(r_{1},\theta_{1})$ and $M_{2}(r_{2},\theta_{2})$ and having the smallest moment of inertia relative to the point $O$. Find the continuous and discontinuous solutions.

Among the lines lying in the upper half-plane and joining the points $A$ and $B$, find the one that, when rotated about $OX$, gives the smallest surface.

Find the discontinuous solutions (see problem 5988).

Find the curve, lying in the region $y \geq k > 0$, joining the points $M_{1}(-a,b)$ and $M_{2}(a,b)$ and forming, when rotated about $OX$, the smallest surface.

Among the lines joining the given points $A(x_{0},y_{0})$ and $B(x_{1},y_{1})$, find the discontinuous solution of the problem of the minimum of the integral

Find the shortest path from the point $A(-2,3)$ to the point $B(2,3)$, lying in the region $y \leq x^{2}$.

Among the curves joining the points $A(0,0)$ and $B(b,0)$ and lying in the region $y \geq 0$, find the one that gives the maximum of the integral

under the condition $b \geq 2a > 0$.

The minimum is sought of the integral

in the region $y \geq f(x)$, under the condition that at one limit the values of the function and its derivative $(a, y_{a}, y'_{a})$ are given, while at the other limit only the number $b$ itself is given. Show that if the extremal meets the boundary curve $y=f(x)$, then its further continuation makes with the boundary curve the same angle, i.e., the angle of incidence equals the angle of reflection.

Inside a convex region $B$ two points $M_{1}$ and $M_{2}$ are given. Find the shortest path from $M_{1}$ to $M_{2}$ that touches the boundary of $B$.

Find the minimum of the integral

under the conditions $y \geq 0$; $1-y'^{2} \geq 0$.

Among the curves joining the points $A(a,y_{a})$ and $B(b,y_{b})$, find the one that gives an extremal value of the integral

under the condition $y \geq 0$, $1-y^{2}y'^{2} \geq 0$.

On the surface $z = \varphi (x,y)$ the points $M_{1}$ and $M_{2}$ are given.

Draw on the surface the shortest line from $M_{1}$ to $M_{2}$, under the condition that the slope of the curve, i.e., $\frac{dz}{ds}$, does not exceed a given number $k$ (mountain railway).

Join the points $A$ and $B$ by the shortest path, leaving $A$ in a given direction and having curvature not exceeding a given value $\frac{1}{R}$. The point $B$ lies outside the circles of radius $R$ tangent to the given direction at the point $A$.

Light travels along a path such that the first variation of the integral

along the path equals zero. Here $n(x,y)$ is the refractive index of the medium.

If a ray passes from one medium into another, then $n(x,y)$ may undergo a sudden change. Derive the condition for the refraction of a light ray when the ray passes from a medium with refractive index $n_{1}$ into a medium with refractive index $n_{2}$ (see problem 6055).