By Russak I.B.

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4. A geodesic on a given surface is a curve, lying on that surface, along which distance between two points is as small as possible. On a plane, a geodesic is a straight line. Determine equations of geodesics on the following surfaces: 2 2 2 2 a. Right circular cylinder. [Take ds = a dθ + dz and minimize or a2 dθ dz a2 + dz dθ 2 dθ 2 + 1 dz] b. Right circular cone. ] c. Sphere. ] d. Surface of revolution. [Write x = r cos θ, y = r sin θ, z = f (r). ] 34 5. Determine the stationary function associated with the integral 1 I = 0 2 (y ) f (x) ds when y(0) = 0 and y(1) = 1, where f (x) = −1 0 ≤ x < 1 1 4 1 4

Only the end conditions y(0) = 0 is preassigned. c. Only the end conditions y(1) = 1 is preassigned. d. No end conditions are preassigned. 8. Determine the natural boundary conditions associated with the determination of extremals in each of the cases considered in Problem 1 of Chapter 3. 9. Find the curves for which the functional I = x1 √ 0 1+y2 dx y with y(0) = 0 can have extrema, if a. The point (x1 , y1 ) can vary along the line y = x − 5. b. The point (x1 , y1) can vary along the circle (x − 9)2 + y 2 = 9.

For the composite arc y13 + G35 + y56 would in that case have the same length as y12 and the arc y13 + L35 + y56 formed with the straight line segment L35 , would be shorter than y12 . It follows then that: If an arc y12 intersecting the curve N at the point 2 is to be the shortest joining 1 with N it must be a straight line perpendicular to N at the point 2 and having on it no contact point with the evolute G of N. Our main purpose in this section was to obtain the straight line condition and also the perpendicularity condition at N for the minimizing arc as we have done above.