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Math Help - Minimizing a functional definite integral

  1. #1
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    Minimizing a functional definite integral

    I have a definite integral defined by

    T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}  G(g)\mathrm{d}g

    where G is a continuous function of a variable g, and g_{1} and g_{2} are known numbers. I want to minimize T\left(G\left(g\right)\right), that is I want to find a continuous function G=f\left(g\right) that makes T\left(G\left(g\right)\right) minimum. Ideally I would differentiate it and equate to zero, but because T\left(G\left(g\right)\right) is too complicated to be obtained and then differentiated analytically, I would like to know if there is a numeric technique or any other technique by which this problem can be solved.
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  2. #2
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    Re: Minimizing a functional definite integral

    Hey James4321.

    Try differentiating both sides and re-arrange to get a DE (not necessarily a nice one) and then use a numerical integration scheme.
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  3. #3
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    Re: Minimizing a functional definite integral

    Your question falls within the subject "Calculus of Variations", and there is only one thing that I know about it: the Euler–Lagrange equation, which you can read about here:

    Euler

    Hope that helps....

    - Hollywood

    P.S. the link text should be "Euler-Lagrange equation". If you know how to fix it, please let me know.
    Last edited by hollywood; December 20th 2012 at 07:25 PM.
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