Good References for Functional Derivatives

[Alamino]

Does anyone know good online references about functional derivatives? Most of the documents contain only the definition, but I would like some more complete material containing, for example, rules for differentiate composite functionals and other details.

 

[ReyChiquito]

You should be looking for Gateaux or Frechet derivatives, its rather easy in the case of integral functionals, which are the most common situation in physics and variational problems. What you are mostly looking is to linearize your functional, the linear part corresponds to the Frechet derivative. For example, let

$$F(x,f(x))=\int_a^b\sqrt{1+f'^2(x)}dx$$

$f(a)=A,f(b)=B$

(path length between A and B)

In this case we can calculate the Gateaux derivative using the definition:

$$D_hf(x)=\frac{\partial}{\partial t}F(x,f(x)+th(x))\mid_{t=0}$$

doing a little algebra you can easyly prove that

$$D_hf(x)=\int_a^b\frac{f'(x)h'(x)}{\sqrt{1+f'^2(x)}}dx$$

The Frechet derivative is more powerfull though, because it allows us to calculate the derivative of more types of functionals.

Let $J(y)$ be a functional of $y(x)$. The Frechet derivative is given by

$$J(y_0+h)=J(y_0)+DJ(y_0)h+o(\|h\|)$$

let

$$J(y)=\int_a^b G(x,y,y')dx$$

then

$$J(y+h)-J(y)=\int_a^bG_y(x,y,y')h+G_{y'}(x,y,y')h'dx+o(\|h+h'\|)$$

$$D_hG(x,y,y')=\int_a^bG_yh+G_{y'}h'dx$$

the example above yields the same result, as expected.

For further references on Gateaux and Frechet derivatives you should check a book of Variational Calculus, i could recommend the following texts.

TROUTMAN. Variational Calculus with elementary convexity.
COURANT. Calculus of Variations.
GELFAND & FOMIN. Calculus of Variations
CARATHEODORY. Calculus of variations and PDE's.
MIKLIN. Variational methods in mathematical physics.
COURANT & HILBERT. Methods of mathematical physics, Vol.I.

 

[Alamino]

Thanks. I appreciate your reply, but what I was looking for was something more general: rules about differentiation of a general functional.

Luckily I could finally find a document that has what I was looking for. For anyone interested, it is at the address


http://phys.cts.nthu.edu.tw/member/staff/qft.pdf

 

[ReyChiquito]

Heh, its the same thing. They use the Frechet derivative. The other results are just properties of it :P

 

[Alamino]

Hmm... True. But you must forgive me, I´m learning it now and I´m not too much used to the details. Anyway, thanks again.

 

[Bruno5antos]

Sorry but i need help in this too...the link doesn't work ...any one has that pdf?

 

[amaresh.tifr]

Thanks. I appreciate your reply, but what I was looking for was something more general: rules about differentiation of a general functional.

Luckily I could finally find a document that has what I was looking for. For anyone interested, it is at the address


http://phys.cts.nthu.edu.tw/member/staff/qft.pdf

the link doesnot work, do u still have the pdf...?