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A functional that depends on an integral?

  1. Mar 23, 2008 #1
    Is it possible to find the extrema of an integral equation if the integral depends on a variable and an integral of that variable, i.e. the integrand is f(x) * g(integral(x)).

    I'm not sure if this is a "nonlocal" functional, or not a functional at all, but I can't find any references that deal with this type of problem.

    If anybody has any ideas of where I could find more information about this, please let me know.
     
  2. jcsd
  3. Mar 25, 2008 #2
    In case the question wasn't clear, here it is in a different way:

    Does the calculus of variations apply to situations where you're looking for some function, y(x), that extremizes (in my particular case)

    [tex]\int{ f(y,x) \ln{ \left[ \int{ f(y,x) dx } \right] } dx }[/tex]

    ?
     
  4. Mar 25, 2008 #3
    I tried to apply the quick-and-dirty physicist's derivation of the Euler-Lagrange equations to this case but came up empty handed. Maybe it is possible to solve for the particular case of the function f you are interested in, but otherwise I am at a loss for how to proceed.
     
  5. Mar 25, 2008 #4
    Maybe I'm being stupid but isn't that equal to

    [tex]\int{ f(y,x) \ln{ \left[ \int{ f(y,x') dx' } \right] } dx }=\ln{ \left[ \int{ f(y,x') dx' } \right]\int{ f(y,x) } dx }=\ln{ \left[ \int{ f(y,x) dx } \right]\int{ f(y,x) } dx }[/tex]

    ...?
     
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