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How to find the function for which ∫ √x*√(1+y'^2) dx is stationary?

  1. Dec 8, 2013 #1
    • This was originally posted in a non-homework forum and does not use the template.
    this is an euler lagrange equation problem from the book- "classical mechanics-John R. Taylor", problem-6.11

    find the path function for which ∫ √x*√(1+y'^2) dx is stationary.

    the answer is- x= C+(y-D)^2/4C, the equation of a parabola.

    here the euler lagrange equation will work on f=√x*√(1+y'^2).

    since ∂f/∂y= 0, so ∂f/∂y'= const

    → √x*y'/ √(1+y'^2)= constant.

    then i dont get how i get from here to the equation of the parabola.

    any help?
     
  2. jcsd
  3. Dec 8, 2013 #2
    Square both sides of the equation, then do algebra till you get ##y'## on one side, and ##x## on the other. Integrate.
     
  4. Dec 9, 2013 #3
    well, squaring gives me- x*y'^2=C(1+y'^2). i could separate to get x=C(1+y'^2)/y'^2

    how do i integrate the terms involving y'^2 ?
     
  5. Dec 9, 2013 #4
    You have not simplified it enough. You can transform that to an equation that does not involve fractions.
     
  6. Jan 8, 2015 #5
    Does anyone know why the problem asks for y=y(x) but the solution is in the form x=x(y)?
     
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