A functional that depends on an integral?

  • Thread starter jfitz
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12
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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.
 
12
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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]

?
 
1,221
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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.
 
[tex]\int{ f(y,x) \ln{ \left[ \int{ f(y,x) dx } \right] } dx }[/tex]

?
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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