1. (4 points) Find the extremals of the functional
subject to the conditions
2. (for practice) Given two points A and B in the xy-plane, let γ be a curve joining them. Among all such curves of given length `, find the one such that γ and the line joining A and B encloses the greatest area. You may assume that A is the origin is B is a point in the first quadrant for simplicity.
3. (for practice) Among all triangles with base fixed at the points (−a,0),(0,a),a > 0 and a given perimeter, which one encloses the greatest area. Try to solve the problem without calculus of variations and then with calculus of variations.
4. (4 points)Among all curves joining a given point (0,b),b > 0 on the y-axis and a point (a,0),a > 0 on the x-axis, and enclosing a given area S together with the x-axis, find the curve which generates the least surface area of solid of revolution when rotated about the x-axis. Full credit if you can write down the functional and constraint and the Euler’s equation. No need to solve it.
5. (4 points) Find the extremal of the functional
satisfying the boundary conditions y(0) = 1,y(1) = 4.
(Hint. The Euler’s equation is . Let . Show that q satisfies .
From this deduce that y(x) = C1(x + C2)2.)
6. (4 points)According to problem 10 on page 130 of the textbook, there are two extremals to problem 5 above. Write down the second variations, δ2J[h], for each of these extremals. Which one of them corresponds to a weak minimum?
