Homework #4
Assigned: Jan. 25th, 2007
Due: Feb. 1st, 2007
Points: 20

1. 10 points
   
2. Compute the Gateaux variation of the functional involved in the brachistochrone problem using the operationally useful definition of Gateaux variation. Write down the differential equation that governs the solution.
3. 10 points
   The lowest eigenvalue of a vibrating string under load q(x) is given by the minimum value of the Rayleigh quotient, R(w). It is given by N/D where
   N = int(0 to L) {E*(w')^2 - q*w^2} dx
   D = int(0 to L) {rho*w^2} dx Note that E is Young's modulus, rho is the mass density per unit length, L is the length of the string, and w(x) is the transverse displacement of the string. dw/dx is indicated as w'. Write down the variation of R(w) with respect to w and obtain the differential equation.