How do we obtain a Taylor expansion of a non-linear functional?

[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.
In another paper (in french) skip to page 9
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.

 

[S.G. Janssens]

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.

 

[stevendaryl]

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.
In another paper (in french) skip to page 9
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.

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.

 

[naima]

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

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

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))