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Euler - Lagrange Equation(changing variable)

  1. Jun 1, 2013 #1
    Create the Euler-Lagrange equation for the following questions (if it's necessary change the variables).
    1. The problem statement, all variables and given/known data


    $$\tag{1}\int _{y_{1}}^{y2}\dfrac {x^{'}{2}} {\sqrt {x^{'}{2}+x^{2}}}dy$$

    $$\tag{2}\int _{x_{1}}^{x_{2}}y^{3/2}ds $$

    $$\tag{3} \int \dfrac {y.y'} {1+yy{'}}dx $$


    2. Relevant equations

    $$ \dfrac {d} {dy}\left( \dfrac {\partial F} {\partial x^{'}}\right) -\dfrac {\partial F} {\partial x}=0 $$

    3. The attempt at a solution
    I don't have an idea about 1 and 3.But here it's what I have tried for 2.

    2) $$\int _{x_{1}}^{x_{2}}y^{3/2}ds = \int _{x_{1}}^{x_{2}}y^{3/2}(1+y'^{2})^{1/2}= \int _{y_{1}}^{y2}\ (1+x^{'2}) (y^{3/2})dy $$
    So our Euler equation is;
    $$ \dfrac {d} {dy}\left( \dfrac {\partial F} {\partial x^{'}}\right) -\dfrac {\partial F} {\partial x}=0 $$

    *Then I have to find Y' or X'.But I did not take a differential equations course yet,we use Beltrami identity to calculate the extremum points.
     
  2. jcsd
  3. Jun 2, 2013 #2

    haruspex

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    You dropped a sqrt: $$F(y, x, x') = (1+x'^2) ^\frac12 y^\frac32 $$
    So what do you get when you substitute for F?
     
  4. Jun 17, 2013 #3
  5. Jun 17, 2013 #4
    I think it will be ;

    $$F(y, x, x') = (1+x'^2)/x' ^\frac12 y^\frac32 $$
     
  6. Jun 18, 2013 #5

    haruspex

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    How do you get that? Isn't F the function in the integral wrt y?
     
  7. Jun 18, 2013 #6
    Before that,I would like to ask can I use the method what I've linked for solve these problems?
     
  8. Jun 18, 2013 #7

    haruspex

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    Looks ok to me. It does get confusing the way you switch between the ∫f(x, y, y')dx form and the ∫f(y, x, x')dy form. Pls fix on one.
     
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