# Minimizing a functional definite integral

• Dec 19th 2012, 12:18 PM
James4321
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.
• Dec 19th 2012, 08:10 PM
chiro
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.
• Dec 20th 2012, 07:06 PM
hollywood
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.