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Math Help - Euler-Lagrange Equation

  1. #1
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    Euler-Lagrange Equation

    Let f = \frac{\dot{x}^2}{t^2}

    The Euler-Lagrange Equation is

    \frac{\partial f}{\partial x} - \frac{d}{dt} \Big(\frac{\partial f}{\partial \dot{x}} \Big) = 0

    I get

    0 - \frac{d}{dt} \Big(\frac{2 \dot{x}}{t^2} \Big) = 0....(Eqn A)

    \frac{2 \ddot{x}}{t^2} = c

    \ddot{x} = \frac{1}{2}ct^2, \dot{x} = \frac{1}{6} ct^3, x = \frac{1}{24}ct^4 + k

    but this is wrong. According to solutions

    x = \frac{1}{3}ct^3 + k where k is a constant.

    I can't see where I have gone wrong from eqn A.
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  2. #2
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    Quote Originally Posted by MrJack1990 View Post
    Let f = \frac{\dot{x}^2}{t^2}

    The Euler-Lagrange Equation is

    \frac{\partial f}{\partial x} - \frac{d}{dt} \Big(\frac{\partial f}{\partial \dot{x}} \Big) = 0

    I get

    0 - \frac{d}{dt} \Big(\frac{2 \dot{x}}{t^2} \Big) = 0....(Eqn A)

    \frac{2 \ddot{x}}{t^2} = c

    \ddot{x} = \frac{1}{2}ct^2, \dot{x} = \frac{1}{6} ct^3, x = \frac{1}{24}ct^4 + k

    but this is wrong. According to solutions

    x = \frac{1}{3}ct^3 + k where k is a constant.

    I can't see where I have gone wrong from eqn A.
    Integrate both sides wrt t: \displaystyle \frac{2 \dot{x}}{t^2} = A \Rightarrow \dot{x} = Bt^2 etc.
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  3. #3
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    \frac{2 \ddot{x}}{t^2} = c
    Should this be

    \frac{2 \dot{x}}{t^2} = c
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  4. #4
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    Quote Originally Posted by MrJack1990 View Post
    Should this be

    \frac{2 \dot{x}}{t^2} = c
    Well yes (why would it not?).
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