Homework #6
Assigned: Feb. 14th, 2007
Due: Feb. 22nd, 2007
Points: 30

1. 10 points
   Consider the functional J(f(t)) = int(t0 to t1){sqrt(f^2 + f^'^2)}dt where f^' indicate (df/dt). Write the E-L equation and obtain the solution of the differential equation. Write the necessary boundary conditions to solve the unknown constants in the solution of the differential equation. Do you see here a hidden problem that has a nice physical (or geometrical) significance?
2. 10 points
   The potential energy (PE) of a plate under transverse load p(x,y) is given as follows.
   PE = double int(over region R) of {D*(uxx+uyy)^2 - 2*D*(1-nu)*(uxx*uyy-uxy^2) - p*u} dx dy
   where D is the plate modulus and nu is the Poisson's ratio. D and nu are known; u(x,y) is the transverse displacement. uxx = second partial derivarive of u w.r.t x and so on for uyy and uxy.
   Minimize PE w.r.t. u and obtain the differential equation that governs the static equilibrium of the deformation of the plate. Extra credit for those who write the boundary conditions and interpret them for different types of supports for a plate.
3. 10 points
   Consider the functional J(y(x)) = int(0 to 10){y'^3}dx where y' = dy/dx, and its boundary conditions y(0) = 0, y(10) = 0. Find the curve(s) for which this functional has extremum value such that the curve(s) do not lie inside the circle given by (x-5)^2 + y^2 = 9. Draw the curves and the circle to show you solutions and attach the plot with your solution.