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Homework Help: Problem of brachistochrone

  1. Mar 23, 2005 #1
    I am gathering my mechanics notes and I put into it some examples. When I get the Hamilton principle I put a section for some basic variation calculus. There's the problem of brachistochrone, I try to solve it, but I get stuck with a integral:

    the integral that I should make minimal is (I'm so sorry, but I don't know how to put it in LaTex):

    t=int((sqrt(1+((dy/dx)^2))/sqrt(2gy))dx)

    and from the variation calculus, the y must be the one that complies:

    df/dy-(df/(dy/dx))/dx=0 where f=(sqrt(1+((dy/dx)^2))/sqrt(2gy)).

    then calculating the partial derivatives of f and putting them into the eq:

    m(1+((dy/dx)^2))+2y(dy/dx)(dm/dx)=0 or
    m(1+(y'^2))+2yy'(dm/dx)

    where y'=dy/dx and m=(1/f)(y'/(gy))

    The question is how to integrate it? or which change of variable is good to integrate it in order to get the eq of the brachistochrone?

    NOTE: I have tried to separate variables, but this is impossible.
     
  2. jcsd
  3. Mar 23, 2005 #2

    Andrew Mason

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    Science Advisor
    Homework Helper

    Bernoulli posed this problem in the late 17th C and Newton solved it but he took 12 hours to do it. And he invented Calculus. Mind you, he did not have the benefit of the Euler-Lagrange approach. Just so you don't drive yourself crazy, a complete solution can be found here:
    http://mathworld.wolfram.com/BrachistochroneProblem.html

    AM
     
  4. Mar 23, 2005 #3
    thank you so much.
     
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