• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,741 comments
1,470 users with positive rep
818 active unimported users
More ...

  Minimizing a functional definite integral

+ 3 like - 0 dislike

I have a definite integral defined by


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.

This post imported from StackExchange Mathematics at 2014-06-02 20:31 (UCT), posted by SE-user James White
asked Dec 19, 2012 in Mathematics by James White (15 points) [ no revision ]
I don't completely understand your problem. Your functional doesn't look "complicated" at all - it maps a function $G$ to it's integral $\int_1^2 G$. However, the 'minimisation' isn't well-defined. If you take $G(g) \equiv c \equiv \text{constant}$, you can make $T(G)$ any value you want, so what would be the minimum? Maybe I'm completely misreading the question, but in that case many others are probably having the same problem.

This post imported from StackExchange Mathematics at 2014-06-02 20:31 (UCT), posted by SE-user Vibert
Look up calculus of variations (en.wikipedia.org/wiki/Calculus_of_variations), to see how these type of problems are normally dealt with. In your case, as Vibert points out, either $f(g)=0$ or $f(g) = -\inf$, depending on how you define minimisation, is the solution to your problem.

This post imported from StackExchange Mathematics at 2014-06-02 20:31 (UCT), posted by SE-user Jaime
Also note that $g$ of the right hand side is swallowed, and this $T$ doesn't depend on $g$ (but on $G$ and $g_1,g_2$).

This post imported from StackExchange Mathematics at 2014-06-02 20:31 (UCT), posted by SE-user Berci

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights