Parametric Form of Integrals, Transversality

Find the maximum of the integral

Find the geodesic lines on the surface of revolution $r^{2} = 2f(z)$ ($r$, $\varphi$, $z$ are cylindrical coordinates). Determine for which arcs $M_{1}M_{2}$ the integral $\int_{M_{1}}^{M_{2}} ds$ is indeed a minimum.

Find the geodesic lines of the circular cylinder $r = R$.

Find the geodesic lines of the cone $r = z \operatorname {tg} \alpha$.

Investigate the geodesic lines on a sphere of radius $R$, find the extremals, investigate Jacobi's condition, and determine the character of the minimum of the integral $\int ds$.

Find the geodesic lines on the paraboloid of revolution $2z = r^{2}$. Under what condition will the geodesic line between two points on the same meridian be the shortest distance?

For the oblate ellipsoid of revolution

the equator ($y=0$, $x^{2}+z^{2}=a^{2}$) is a geodesic line. Show that the critical point $A'$ conjugate to the point $A$ on this line is at a distance $\pi b$ from $A$.

We have two circles of radius $r$ with centers at the points $M_{1}(-a,b)$ and $M_{2}(a,b)$. A curve lies in the upper half-plane, ending on the given circles, and when rotated about $OX$ gives a surface of the smallest area. At what points and at what angle does it intersect the circles?

Write down the transversality condition for the integral

For which integrals of the form $\int F(x,y,y') \, dx$ does transversality reduce to orthogonality?

Find all integrals $\int f(x,y,y') \, dx$ for which the transversals intersect the extremals at an angle of $45°$.

Among the lines joining the point $O(0,0)$ with the curve $y^{3} = 2-x$, find the one that gives the minimum of the integral

In the vertical plane $XOY$ with the axis $OY$ directed downward, find the curve, moving along which a heavy material point will arrive in the shortest time from the point $O$ to a given line $y = (x-c)\operatorname {tg}\beta$; $0 < \beta < \pi$. Here the initial velocity is zero.