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Mechanical variation involving auxiliary functions

  1. Jan 24, 2016 #1
    1. The problem statement, all variables and given/known data
    A chain of length L and uniform mass per unit length ρ is suspended in a uniform gravitational field. The potential energy U[y] and length l[y] functionals of the chain can be written in terms of y(x) as follows:

    U[y] = ρg*Int(y(1+y'^2)^1/2 dx) l[y] = Int((1+y'^2)^1/2) dx

    where (x = 0, y = 0) and (x = a, y = 0) are the end points of the chain. Deduce the shape of the chain using the following procedure:
    Everything after this point are guidelines.
    - Write down an appropriate auxiliary functional for the problem (i.e. a functional that includes a Lagrange multiplier). - Find the first integral E(y, y′ ) associated with the auxiliary functional. - Use the first integral to find y ′ in terms of y, the Lagrange multiplier and the conserved quantity corresponding to the first integral. - Find the general solution to the first-order ordinary differential equation for y(x) found in the previous step. Your answer should contain two constants of integration (one of them will be the conserved quantity). - Use the boundary conditions to eliminate the second constant of integration and the Lagrange multiplier. - Calculate l[y] and find the length L of the chain in terms of the physical parameters ρ, g, a and the conserved quantity

    2. Relevant equations
    Sλ[q(t)] = S[q(t)] − λF[q(t)]

    3. The attempt at a solution
    Well i tried to create an auxiliary equation in which my alpha(y,y') = pgy*(1+y'^2)^1/2 - λ(1+y'^2)^1/2

    Which i am 100% sure to be wrong, as when i follow through I get to the point where I get everything cancelled out and am left with 1 = 0.
    If someone were to provide me with a corrected auxiliary function i believe i should be able to follow the instructions. But if you would like to talk me through the rest of the question i would appreciate it.
     
  2. jcsd
  3. Jan 30, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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