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Math Help - Variational problems in two variables

  1. #1
    Super Member Showcase_22's Avatar
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    Variational problems in two variables

    Write down the Euler-Lagrane equations for critical points of the functional I of two functions x(t), \ y(t) where:

    I(x,y)=\int_{t_1}^{t_2}(x^2+y^2)\sqrt{1+(x')^2+(y'  )^2}
    I'm slowly working my way through this sheet, but I keep encountering things not in the notes. I'm also having trouble understanding what wikipedia has to say: Euler?Lagrange equation - Wikipedia, the free encyclopedia

    I have a function f(x,y,y',x')=(x^2+y^2)\sqrt{1+(x')^2+(y')^2} so my first equation is:

    \frac{\partial f}{ \partial y}-\frac{\partial}{\partial x} \left( \frac{\partial f}{\frac{\partial y}{\partial x}} \right)- \frac{\partial}{x'} \left( \frac{\partial f}{\frac{\partial y}{\partial x'}} \right)

    The second equation would be the same but with y' instead of x'.

    Am I reading this correctly?
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  2. #2
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    Quote Originally Posted by Showcase_22 View Post
    I'm slowly working my way through this sheet, but I keep encountering things not in the notes. I'm also having trouble understanding what wikipedia has to say: Euler?Lagrange equation - Wikipedia, the free encyclopedia

    I have a function f(x,y,y',x')=(x^2+y^2)\sqrt{1+(x')^2+(y')^2} so my first equation is:

    \frac{\partial f}{ \partial y}-\frac{\partial}{\partial x} \left( \frac{\partial f}{\frac{\partial y}{\partial x}} \right)- \frac{\partial}{x'} \left( \frac{\partial f}{\frac{\partial y}{\partial x'}} \right)

    The second equation would be the same but with y' instead of x'.

    Am I reading this correctly?
    It might be clearer if we write it as

     <br />
\frac{\partial L}{\partial x} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) = 0<br />

     <br />
\frac{\partial L}{\partial y} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{y}}\right) = 0.<br />
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