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Homework Help: Calculus of variations

  1. Oct 28, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the extremal for the case
    [tex]\int_a^b y^2(1+(y')^2) \, dx[/tex]
    where [itex]y(a)=y_{0}, y(b)=y_{1}[/itex]

    2. Relevant equations

    3. The attempt at a solution
    Using the Euler-Lagrange equation for a functional that doesnt depend on x I get
    [tex]F-y'\frac{\partial F}{\partial y'}=c[/tex]
    [tex]\Leftrightarrow y^2(1-(y')^2)=c[/tex]
    [tex]\Leftrightarrow \int \frac{1}{\sqrt{1-\frac{c}{y^2}}}dy=\int dx[/tex]
    [tex]\Leftrightarrow y=\frac{-c}{x^2-1}[/tex]
    Now I have to sub this y(x) into the original integral and I am comfortable doing the integral apart from what to do for the upper and lower limits of integration.
  2. jcsd
  3. Oct 28, 2014 #2


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    2017 Award

    Staff: Mentor

    There should be another free parameter from the integration.
    You can use the limits of the integration to fix those two parameters.
  4. Oct 28, 2014 #3


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    Science Advisor
    Homework Helper

    That doesn't look like any version of Euler Lagrange I know. I'd try looking it up again. You should get a 2nd order ODE.
  5. Oct 28, 2014 #4
    In this case F does not depend on x so the E-L equation is reduced to
    [tex]F-y'\frac{\partial F}{\partial y'}=c[/tex] and you are left with a 1st order ode.
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