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Calculus of Variations Euler-Lagrange Diff. Eq.

  1. Apr 9, 2009 #1
    I'm in dire need of help in understanding calculus of variations. My professor uses the Mathews and Walker text, second edition, entitled Mathematical Methods of Physics and, he has a tendency to skip around from chapters found towards the beginning of the text to those nearer the end. I couldn't say if that was an intended property of this text but, anything beyond freshman level kinematics or electricity and magnetism is only offered on a two-year rotation at the college which I attend so, my exposure to potential prerequisites for the subject matter covered in the course is spotty, at best.

    The text asks that I consider a function:

    F(y, dy/dx, x) and the integral I = [tex]\int[/tex] F(y, dy/dx, x) dx, evaluated from a to b = I[y(x)]

    Then, the text indicates that the objective would be to choose the function y(x) such that I[y(x)] is either a maximum or a minimum...("or (more generally) staionary.")

    It continues:

    "That is, we want to find a y(x) such that if we replace y(x) by y(x) + [tex]\xi[/tex](x), I is unchanged to order [tex]\xi[/tex], provided [tex]\xi[/tex] is sufficiently small.

    In order to reduce this problem to the familiar one of making an ordinary function stationary, consider the replacement
    y(x) y(x) + [tex]\alpha\eta[/tex](x)
    where [tex]\alpha[/tex] is small and [tex]\eta[/tex](x) arbitrary. If I[y(x)] is to be stationary, then we must have
    dI/d[tex]\alpha[/tex], evaluated at [tex]\alpha[/tex]=0, = 0
    for all [tex]\eta[/tex](x)."

    Could someone offer a "dumbed-down" explanation of what the text attempts to communicate?
     
  2. jcsd
  3. Apr 9, 2009 #2
    I'm not sure that I truly understand what it means to have a function like:

    F(y, dy/dx, x). This is a function dependent upon not only x and y but, upon dy/dx as well? If that's correct, what does that mean exactly? In what scenario would one see something like that? Honestly, I'm probably doing pretty well to truly understand the correlation between x and y when y is a function of x: y(x).

    Thank you.
     
  4. Apr 9, 2009 #3

    tiny-tim

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    Hi avocadogirl! :smile:
    For example, the energy of a body might be 1/2 mv2 + mgh + Be-kt

    that's a function of dh/dt and h and t separately :wink:

    (the point is that it's the way F is written that matters … once you solve the equation, you could presumably just write F = G(h,t) or even F = H(t) … but so long as it's still written F(dh/dt, h,t) it can be differentiated separately with respect to each of the three variables)
     
  5. Apr 10, 2009 #4
    Thank you. That does make more sense, especially when thinking of the function in such a way where the components might be differentiable.

    Could someone elaborate a little about the paragraph in the text, in its entirety?

    Thank you, sincerely.
     
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