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Variational Calculus - Minimal Arc lenght for given surface

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data
    A curve is enclosing constant area P. By means of variational calculus show, that the curve with minimal arc length is a circle,


    2. Relevant equations



    3. The attempt at a solution

    [tex]F(t)= \int_{x_{1}}^{x_{2}}\sqrt{1+(f^{'})^{2}}dt[/tex]
    [tex]G(t)= \int_{x_{1}}^{x_{2}}f(t)=const[/tex]

    If i use Lagrange theorem for functionals i will get

    [tex]\int_{x_{1}}^{x_{2}}=\sqrt{1+(f^{'})^{2}} +\lambda f dx [/tex]

    Since, upper functional is not a function of T i can use Euler-Lagrange equation, after differentiation solution will be:
    [tex]1+(f^{'})^{2}=\frac{1}{(C-\lambda f)^{2}}[/tex]
    What should i do now ?
     
  2. jcsd
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