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Homework Help: Missing step: Euler-Lagrange equations for the action integral

  1. Dec 4, 2005 #1
    Hi its me again, stuck once more. Sorry guys and gals :P
    Ok a problem I found on http://en.wikipedia.org/wiki/Action_(physics)
    In a 1-D case how do we get from:
    [tex]\delta S = \int_{t_1}^{t_2} [L(x + \varepsilon, \dot{x} + \dot{\varepsilon})-L(x,\dot{x})]dt[/tex]
    to:
    [tex]\delta S = \int_{t_1}^{t_2} \left(\varepsilon \frac{\pd L}{\pd x} + \dot{\varepsilon} \frac {\pd L} {\pd \dot{x}}\right)dt[/tex]
    where [tex] \varepsilon = x_1(t) - x(t) [/tex]
    and where the first order expansion of L in ε and ε′ is used? I dont even know what that last phrase means, so if someone could explain that to me too, that would be great.
    Thankyou very much.
     
    Last edited: Dec 4, 2005
  2. jcsd
  3. Dec 5, 2005 #2

    George Jones

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    Consider a real-valued function of 2 variables, say [tex]f = f(x,y)[/tex]. Taylor's theorem for multivariable calculus gives

    [tex]
    f(x + \Delta x , y + \Delta y) - f(x,y) = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + ...
    [/tex]

    In your variation principle, [tex]L[/tex] is a function of the "independent variables" [tex]x[/tex] and [tex]\dot{x}[/tex], so

    [tex]
    L(x + \varepsilon , \dot{x} + \dot{\varepsilon}) - L(x,\dot{x}) = \frac{\partial L}{\partial x} \varepsilon + \frac{\partial L}{\partial \dot{x}} \dot{\varepsilon} + ...
    [/tex]

    Neglecting the ... (i.e., the terms higher than first-order) gives the desired result.

    Note: the "independent variables" actually are functions themselves.

    Regards,
    George
     
    Last edited: Dec 5, 2005
  4. Dec 5, 2005 #3
    Thanks George! =)

    Edit: just posting a version of the last formula that shows correctly;

    [tex]L (x + \varepsilon , \dot{x} + \dot{\varepsilon}) - L (x, \dot{x}) = \frac{\partial L}{\partial x} \varepsilon + \frac{\partial L}{\partial \dot{x}} \dot{\varepsilon} + ...[/tex]
     
    Last edited: Dec 5, 2005
  5. Dec 5, 2005 #4
    need more help on this one sorry...

    I still don't understand how to get from one to other. I thought boning up on the Taylor theorem for multiple variables wouldn't be too hard, but I was wrong.
    Can some one post a step by step, dummies guide to getting from:
    [tex]f(x + \Delta x , y + \Delta y) - f(x,y)[/tex]
    to:
    [tex]\frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + ...[/tex]
    please?
    thanx
    38
     
    Last edited: Dec 5, 2005
  6. Dec 5, 2005 #5
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