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Functional Derivatives/Euler-Lagrange

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data
    Hi, I'm working on research and I hit a roadblock with something that should be very simple but I can't solve it because it gets so messy. If anyone can let me know how to do this, it would be greatly appreciated.

    I have a functional T:
    [tex]
    T = \int_{\lambda_{1}}^{\lambda_{2}} sqrt{\sum_{I=1}^{n}} \sum_{i=1}^{d} (\frac{d}{d \lambda}(\sum_{j=1}^{d} s(\lambda) R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda))))^2} d \lambda
    [/tex]

    I need to take functional derivatives with respect to each function defining T and find when they are all concurrently zero. I believe, the Euler-Lagrange equation is able to do this?

    I found what [tex]\frac{\partial{f}}{\partial{g}}[/tex] is, where g is just a place holder for [tex]s(\lambda), R_{j}^{i}(\lambda), a^{j}(\lambda), q_{I}^{j}(\lambda)[/tex]

    Everytime I tried substituting say, df/dR, or df/ds I would get a huge 22-term, with 5 derivatives, impossible equation that I would have to take f with respected to, where [tex]T = \int_{\lambda_{1}}^{\lambda_{2}} f d \lambda.[/tex]

    I'm looking at this euler-lagrange form:
    [tex]\frac{\partial{f}}{\partial{x}} = \frac{d}{d \lambda} (\left \frac{\partial{f}}{\frac{d}{d \lambda}(\frac{\partial{f}}{\partial{x}})} \right) = 0 [/tex]
    Where, each member [tex]s(\lambda), R_{j}^{i}(\lambda), a^{j}(\lambda), q_{I}^{j}(\lambda)[/tex] are in terms of "x".

    Any tips, advice, ideas would be great.
     
  2. jcsd
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