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Calculus of Variations-Hanging String

  1. Nov 20, 2011 #1
    Calculus of Variations--Hanging String

    1. The problem statement, all variables and given/known data

    A uniform string of length 2 meters hangs from two supports of the same height, 1 meter
    apart. By minimizing the potential energy of the string, find the equation describing the
    curve it forms and, in particular, find the vertical distance between the supports and the
    lowest point of the string.

    2. Relevant equations
    U = λ g [itex]\int y(x)dx[/itex]
    0≤ x ≤ 1
    S = [itex]\int\sqrt{dx^2+dy^2}[/itex]dt
    Euler-Lagrange Equation

    3. The attempt at a solution

    Been stuck on this one for awhile, tried solving naively by just solving the lagrange equation for potential energy and that obviously got me an answer of U = 0. I also tried to solve the arc length equation for y(x) and I got y'[x] = √3, so that is obviously wrong. I'm not sure how to add the constraints of the supports into the equation. Im probably going to have to parameterize y and x in terms of some other variable.

    I have a feeling that the shape is going to be a cycloid...just having a hard time proving it.
  2. jcsd
  3. Nov 20, 2011 #2
    Re: Calculus of Variations--Hanging String

    Bump^^^ Can i get a hand with this, i saw a similar question posted, with an unclear answer
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