6
$\begingroup$

In a finite dimensional space, one calculates the Lie derivative as

$L_f(g)(x) = \langle \nabla g, f \rangle$

What is the equivalent in an infinite dimensional space? For example if $g$ takes as argument a function and $f$ is an infinite dimensional vector, how does one think about the Lie derivative?

I am familiar with the Gateau derivative, so does one simply replace the gradient with this? Then we might have

$L_f(g)(x) = \langle dg, f \rangle$

for some appropriate inner product? for instance $L^2$?

$\endgroup$
4
+25
$\begingroup$

You might find what you're looking for in one of the following links:

  1. Lang: http://www.springer.com/gp/book/9780387985930
  2. Kriegl & Michor: http://bookstore.ams.org/surv-53
  3. Choquet-Bruhat & DeWitt-Morette: https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0
  4. Omori: http://bookstore.ams.org/mmono-158/

Basically, consider looking for material on infinite dimensional manifolds.

$\endgroup$
  • 1
    $\begingroup$ Well, thanks, but, I would like some attempt at an answer here. $\endgroup$ – James S. Cook Aug 24 '17 at 21:40
  • 1
    $\begingroup$ Why not just try reading the sections covering the Lie derivative in any of those books? $\endgroup$ – g.s Aug 26 '17 at 5:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.