Homework #5
Assigned: Sep. 1st, 2005
Due: Sep. 8th, 2005
Points: 20

1. 10 points
   Solve the hanging-chain problem with a constraint that the distribution of the mass m is uniform along the horizontal line. That is, (dm/dx) = m^*. The chain hangs between (x₁,y₁) and (x₂,y₂) and has a length of L.

   This problem is given in a book entitled "Calculus of Variations" by Robert Weinstock on page 64 in Chapter 4's exercises. The following is the exact statement of the problem in the book.

   A rope of given length L hangs in equilibrium between two fixed points (x₁,y₁) and (x₂,y₂) in such fashion that the distribution of its mass M is uniform with respect to the horizontal; that is, (dM/dx) = alpha, a given constant, in equilibrium configuration. Show, by means of methods developed in the foregoing chapter, that the shape of the haniging rope must be parabolic. HINT: A certain quick, thoughtless attack upon the problem yields a circular shape; this is of course wrong! A second, swindling approach makes use of equation (24) of 4.2(e) to obtain parabolic shape, but this is likewise wrong. A thoughful approach takes into account the precise nature of the comparison of the potential energy of the rope's equilibrium configuration with other configurations consistent with the constraints; this leads to the required answer
   C₁(y-C₃) = (1/2/alpha) (x-C₂)².

   Equation (24) of 4.2(e) is y' (do F/ do y') - F = Constant where F is the integrand in the functional that depends only on x, y(x), and y'(x).

2. 10 points
   Formulate a calculus of variations problem of the following type, preferably in mechanics or in geometry. The objective functional should depend on two functions y₁(x) and y₂(x). There should be an algebraic constraint in terms of y₁(x) and y₂(x). The constraint function may also involve x.
   Please solve the problem and interpret the boundary conditions and the physical meaning of the Lagrange multiplier function.