B Functional derivative

1. Oct 28, 2016

naima

Hi PF
I try to understand how we get get a Taylor expansion of a non linear functional.
I found this good paper
here F maps functions to scalars. F[f] is defined. It has not scalars as arguments. I agree with A13 and A18.
the fisrt term is $\int dx P_0 (x)$ and all the terms have one more variable.
Do you understand the second point of view? we did not start with densities.

2. Oct 28, 2016

Krylov

Could you explain into some more detail what exactly is confusing you?

In my experience, "functional derivative" usually means what mathematicians call a Gâteaux derivative. Sometimes it refers to the stronger concept of Fréchet derivative. Using the latter notion it is possible to set up a calculus that looks very much identical to the ordinary multivariable calculus for functions defined on open subsets of $\mathbb{R}^n$. For this it is necessary to specify the domain of the nonlinear functional (or operator) in question as a suitable open subset of a normed linear space. Once this is done, familiar theorems such as Taylor's hold almost verbatim.

Now, I know that physicists often do not like to talk about the function spaces that underpin their work, but in this case I believe it really does pay off to break that habit. Unfortunately I cannot read texts that lack rigor (your first link presents itself like that), so for a more rigorous but still gentle reference I would also like to mention "A Primer of Nonlinear Analysis" by Ambrosetti and Prodi.

3. Oct 28, 2016

stevendaryl

Staff Emeritus
I think you mean page 7 of the second paper.

In the second paper, they seem to be assuming that the functional $F[f]$ can be written in the form:

$F[f] = \int dx F(f(x))$

This is a very confusing notation. If you look at the bottom of page 7, you'll see they define:

$S[x(t)] = \int dt L(x, \dot{x})$

which doesn't fit the pattern of $F[f]$. What I think they mean is something like this: They are assuming that the functional $F[f]$ can be written in the form:

$F[f] = \int dx \tilde{F}$

Where the expression $\tilde{F}$ inside the integral can involve $f$ and its derivatives evaluated at the point $x$. Writing this as $F(f(x))$ gives the mistaken impression that it is an ordinary function $F(y)$ evaluated at the point $y=f(x)$.

But in any case, the two papers seem to agree about the functional derivative: Look at equation A.28 in section A.3 in the first paper, and compare it with equation 1.13 page 6 of the second paper. What's confusing about the second paper is that they seem to be making the distinction between

1. $\frac{\delta F[f(y)]}{\delta f(x)}$, and
2. $\frac{\delta F[f]}{\delta f(x)}$
They seem to be treating the second expression as the integral of the first expression:

$\frac{\delta F[f]}{\delta f(x)} = \int dy \frac{\delta F[f(y)]}{\delta f(x)}$

To me, this is an extremely confusing convention. And the author is not even consistent about it, because in equation 1.26 on page 8, they write $\frac{\delta S[x(t)]}{\delta x(t)}$, when it would seem like they should be writing $\frac{\delta S[x]}{\delta x(t)}$ (with no argument $t$ on the function $x$ in the expression $S[x]$). It's very confusing, because it's unclear when they are using $x(t)$ to mean a function, and when they are using it to mean a number, the value at point $t$.

I think it's bad notation, but that the two papers probably mean the same thing.

4. Oct 29, 2016

naima

I found the "trick"
When you have number a and a normalized function G (such that $\int dx G(x)= 1$ you can write $a = \int dx aG(x)$
Here the author expands the functional like in the first paper and then he takes any normalized function G and calls it F(f(x))