1. (5 points) (Brachistochrone problem)In Lecture 11, we derived the functional
which represents the time it takes for a particle traveling on the graph of the function y(x), starting from the point (0,b) to arrive at the point (a,0) under the gravitational force. (a) Using the Beltrami’s identity, find the Euler-Lagrange equation of this functional, (b) Show that
solves your Euler-Lagrange equation. Note that I am not asking you to solve the Euler-Lagrange equation, but just to show that the above is a solution. The function (x(θ),y(θ)) is a parametric curve in the plane for θ ∈ [0,θˆ]. Note that (x(0),y(0)) = (0,b) is automatically satisfied. Use the boundary condition at the other endpoint to obtain the conditions: x(θˆ) = a,y(θˆ) = 0 to find the constants C and θˆ. You don’t need to do this but assume this is done numerically.
2. (5 points) Which curve minimizes the integral
The values of y(x) are not specified at the endpoints.
3. (Bonus, 5 points)A river with parallel straight banks b units apart has stream velocity given by v(x,y) = v(x)j , where j is the unit vector in the y direction. Assume that one side of the bank is the y-axis (x = 0) and the point (0,0) is the point of departure. What route should a boat take to reach the opposite bank (x = b) in the shortest possible time? Assume that the speed of the boat in still water is c, where c > v.
(Hint.
I will derive this functional in class.)
4. (5 points)Find the extremal(s) of the functional
in the class of D2[0,b] functions with y(0) = 0 and y(b) free. Here, 0 < k < 1.
5. (Bonus, 5 points) Find the extremal of the functional
J[y] = ∫ √x2 + y2√1 + (y′)2 dx
(Hint. Use polar coordinates.)
