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Math Help - Calculus of Variations.

  1. #1
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    Calculus of Variations.

    Iím practicing some problems for applied mathematics, and couldnít quite solve the following:
    In calculus of variations, if the speed is x, and the travel time is t = Integral(0 to 1) (1/x)Sqrt(1+(uí)≤)dx with u(0)=0 and u(1)=1, how can see from the Euler equation that uí/x(sqrt(1+(uí)≤)) is constant?
    And how can I integrate once more in order to get the optimal path u(x)?
    If you know the solution, or could guide me in the right direction, I would appreciate it a lot! Thanks in advance..
    Last edited by mr fantastic; April 25th 2010 at 04:30 AM. Reason: Restored deleted question.
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  2. #2
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    Quote Originally Posted by francoiskalff View Post
    Iím practicing some problems for applied mathematics, and couldnít quite solve the following:
    In calculus of variations, if the speed is x, and the travel time is t = Integral(0 to 1) (1/x)Sqrt(1+(uí)≤)dx with u(0)=0 and u(1)=1, how can see from the Euler equation that uí/x(sqrt(1+(uí)≤)) is constant?
    And how can I integrate once more in order to get the optimal path u(x)?
    If you know the solution, or could guide me in the right direction, I would appreciate it a lot! Thanks in advance..
    If \frac{u'}{x\sqrt{1+u'^2}} = c_1

    Square both sides and solve for u'^2 - then solve for u'.
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    Quote Originally Posted by Danny View Post
    If \frac{u'}{x\sqrt{1+u'^2}} = c_1

    Square both sides and solve for u'^2 - then solve for u'.
    Hey thanks for your reply, but I think you misinterpreted my question. I guess I wasn't clear enough. I meant to ask:
    if the speed is x, and the travel time is t = Integral(0 to 1) (1/x)Sqrt(1+(u’)≤)dx with u(0)=0 and u(1)=1, how can you see from the Euler equation WHICH QUANTITITY is constant (Snell's law)?
    I know the answer is:
    \frac{u'}{x\sqrt{1+u'^2}} = c_1 but i dont know how to get there...
    Last edited by mr fantastic; April 25th 2010 at 04:31 AM. Reason: Restored deleted post.
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