# The Frechet derivative and Lagrange multipliers on Banach spaces

I am interested in questions of the following form: minimize $H(f)$ given $G(f) = 0$ where $H$ and $G$ are operators of type $X \to R$ where $X = R \to R$. An example is:

Minimize $$H(f) = \int_{-1}^1\sqrt{1+f'(x)^2}$$

Under the conditions that:

$$G(f) = \int_{-1}^1f(x) = \pi/2$$ $$f(-1) = f(1) = 0$$

That is, find a function f on [-1,1] with area under the curve equal to $\pi/2$, minimizing the path length (the answer to this example is a half-circle $f(x) = \sqrt{1-x^2}$, I believe).

Other examples of operators that can occur in the minimize or in the condition are:

$$A(f) = f(1)$$

$$B(f) = \int_a^b E(x,f(x),f'(x))dx$$

(i.e. an integral of an expression containing $x$, $f(x)$ and $f'(x)$).

The way to solve these kind of problems seem to be Lagrange multipliers on Banach spaces. How does one do this?

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The optimum in the first case looks to me like it should be a constant function. –  Noah Stein Jan 26 '11 at 16:47
Dear Jules, your post could be a bit more easy on the eyes if you use the LaTeX support on this website. –  Willie Wong Jan 26 '11 at 16:49
Hmm yes you are right...how about the problem with the additional constraint f(0) = f(1) = 0? –  Jules Jan 26 '11 at 16:49
It depends what you mean by a solution, and the level of rigour you want. Physicists and engineers solve similar problems non-rigorously all the time, by Calculus of Variations methods (which involve Lagrange multipliers). Euler-Lagrange equations are the keywords to search for. However, they usually don't specify exactly which functions they consider. Banach spaces only arise if your restrictions are explicit: $f$ being $C^1$, $C^2$, etc., and you define a complete norm on the class of functions. But for most specific problems, Banach space theory is probably not worth the effort. –  Zen Harper Jan 26 '11 at 18:27
It seems to me that the calculation part of Calculus of Variations (i.e., how to find the Euler-Lagrange equations), including constrained problems, is presented in many physics, namely classical mechanics, textbooks. –  Deane Yang Jan 26 '11 at 20:33

NOTE: this was a comment, because I thought it wasn't detailed enough for an answer; but Jules (the OP) specifically asked me to post it as an answer.

NOTE to Jules: However, maybe you should wait a few hours or days before accepting any answer, to give others a chance to read it (differing time zones around the world, etc.) [Some MO people seem to get a bit annoyed when people accept answers very quickly].

Sorry I can't think of any particular book to recommend for Calculus of Variations; but I think an advanced undergraduate book (maybe just a chapter or two of a "Mathematical Methods" book) might be more useful than a detailed Graduate level book, which would probably be more than you need.

It depends what you mean by a solution, and the level of rigour you want. Physicists and engineers solve similar problems non-rigorously all the time, by Calculus of Variations methods (which involve Lagrange multipliers). Euler-Lagrange equations are the keywords to search for. However, they usually don't specify exactly which functions they consider. Banach spaces only arise if your restrictions are explicit: being $C^1$, $C^2$, etc., and you define a complete norm on the class of functions. But for most specific problems, Banach space theory is probably not worth the effort.

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