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Math Help - Euler-Lagrange Equations

  1. #1
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    Euler-Lagrange Equations

    I am a bit confused by the following problem, I am comfortable with just one Euler-Lagrange (I think!) equation but this has two and has me a bit confused :

    Solve the following using Euler-Lagrange equations

    J[\vec{x}]=\int_{a}^{b}\frac{\sqrt{(x'_1)^2+(x'_2)^2}}{x_2}d  t

    Thanks for any help

    (Is there any way to get LaTex anymore??)
    n
    So this is what I have so far (didn't post earlier as I couldn't get the LaTex to work)

    The Two Euler-Lagrange equations give:

    1. 0-\frac{d}{dt}(\frac{x'_1}{x_2\sqrt{(x'_1)^2+(x'_2)^  2}})=0

    2. -\frac{\sqrt{(x'_1)^2+(x'_2)^2}}{x_2^2}-\frac{d}{dt}(\frac{x'_2}{x_2\sqrt{(x'_1)^2+(x'_2)^  2}})=0

    So we may conclude that  \frac{x'_1}{x_2\sqrt{(x'_1)^2+(x'_2)^2}}=k and so
     \frac{x'_1}{x_2*k}=\sqrt{(x'_1)^2+(x'_2)^2}

    substituting this into equation 2. we get

     -\frac{x'_1}{x^3_2*k}-\frac{d}{dt}(\frac{x'_2}{x_1'})

    However I have no idea what to do now
    Last edited by hmmmm; May 2nd 2011 at 06:01 AM. Reason: Fixed up LaTeX code/putting in my work do far
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  2. #2
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    One temporary fix is to use [tex] tags instead of [tex] tags.

    -Dan
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  3. #3
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    Just a thought: instead of solving the first integral of the first equation for the square root, why not eliminate x_{1} altogether from the second equation? That is, solve

    \frac{\dot{x}_{1}}{kx_{2}}=\sqrt{ \dot{x}_{1}^{2}+\dot{x}_{2}^{2} }

    as follows:

    \dot{x}_{1}^{2}=\frac{\dot{x}^{2}}{1-k^{2}x_{2}^{2}}.

    [EDIT]: This is incorrect. See below for correct solution.

    Where does that lead?
    Last edited by Ackbeet; May 2nd 2011 at 10:41 AM.
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  4. #4
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    After the above substitution into Equation (2), I found the substitution

    u=\sqrt{\frac{1-k^{2}x_{2}^{2}}{2-k^{2}x_{2}^{2}}}

    useful.
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  5. #5
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    When I solve for  x'_1 I get x'_1^2=\frac{k^2*x_2^2*x'_2^2}{1-k^2x_2^2} ? have I made some simple mistake that I am not noticing?
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  6. #6
    A Plied Mathematician
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    Nope. You're right, and I'm wrong. See where your result leads you.
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  7. #7
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    After I make that substitution I have absolutely no Idea where to go sorry.
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  8. #8
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    Let me back up a minute, here. Are you trying to minimize or maximize J? Are there any constraints on the dependent variables?
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  9. #9
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    My past paper question simply states "find the Euler-Lagrange equations" of what I had above so I presume that I meant to find the extremals (as i have had to do in the past however I have never had the same trouble with them) I cannot see how to solve this and there is a possibility that the question was wrong (and corrected the day of the exam) I am very sorry if this is the case
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  10. #10
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    Well, here's what happened with me: after substituting the first equation into the second, I ended up with 0 = 0. That is, the second equation doesn't give me any more information than the first! So your integrated equation there is about as far as you can go with the EL equations. There is more you could say about the problem, though, from a heuristic point of view. Let's say you're trying to minimize the integral, and both coordinates must be non-negative. Why then, just make both coordinates equal to a nonzero constant, and then J = 0. Since J is always non-negative (under the assumptions I just mentioned), you've found your minimum.

    Without some constraints, I don't think the problem is solvable.

    Cheers.
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  11. #11
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    Ok thanks maybe that is all I have to do as I say my past paper questions are worded quite badly so that is probably all I have to do, hopefully it is clearer for my exam in a week! Thanks very much for your help and sorry about the problem not being very clear.
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  12. #12
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    Quote Originally Posted by hmmmm View Post
    Ok thanks maybe that is all I have to do as I say my past paper questions are worded quite badly so that is probably all I have to do, hopefully it is clearer for my exam in a week! Thanks very much for your help and sorry about the problem not being very clear.
    Hardly your fault! You're welcome for whatever help I could provide.
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