# Talk:Lie derivative

## Motivation needed

Sorry, but this article was obviously written by mathematicians.... It describes exactly what a Lie derivative is, but gives no clue whatsoever why it was defined or its uses. I've been seeing references to Lie derivatives in differential geometry and thought I could get an idea of why I should care to learn about them from Wikipedia, but instead got this dry definition more suitable for computers than humans. Could someone who knows more about the subject add sections on the history of the Lie derivative, its reasons for being singled out for definition by Lie, etc.

I believe this topic has something to do with the intersection between algebra and differential geometry, am I correct?

Well, if you've "been seeing references to Lie derivatives in differential geometry" then surely the setting within which these references arise are themselves motivation! Dharma6662000 (talk) 12:07, 5 August 2008 (UTC)

It seems that in the intro Lie derivate is used both for L and for elements of the range of [-,-]. -MarSch 14:19, 27 Apr 2005 (UTC)

## Where is the sign = ? where is the true ?

(sorry for my very bad English)

The first equation is correct. The second is not. 69.129.194.27 05:23, 22 July 2006 (UTC)

## Using math sections etc.

Hello,

I just edited a bit of the "Lie derivative of tensor fields" section. I made a few clarifications in the first (and very long) sentence, and I converted the math to math-markup. I know this might be controversial, but in this case there is a really long tensorial equation at heart that is somewhat tedious to read as HTML and gains a lot of clarity in the TeXed version. Hope that's OK.

- Thomas, 131.111.231.27

## omega

it seems linas that you have used omega for a 1-form here. This clashes with the supposed standard use of omega as the volume form. Perhaps we should start using ${\displaystyle \mathrm {vol} }$ for the volume form. I think it is less confusing. -MarSch 15:56, 27 Apr 2005 (UTC)

I dunno, ω is used for many things; often it is a representative member of Ωk which is the set of smooth sections of Λk. I'm not too worried; which is which is should always be clear from context. There's not enough letters in the Greek alphabet. If its not clear from context, look for the phrase "where ω is a ...". I've never seen vol used for the volume form, that seems like a rather non-standard notation. BTW, add your book reference to the article on Laplacian. linas 00:33, 29 Apr 2005 (UTC)
I will read it. Why did you remove it? It is from there that my use of the notation stems. It may not be standard, but it is certainly clearer. -MarSch 13:36, 2 May 2005 (UTC)

Has somebody a bot who could fix all links to derivation by adding (abstract algebra)? MFH: Talk 22:20, 10 May 2005 (UTC)

ask User:Oleg Alexandrov, he has a bot. -MarSch 14:43, 11 May 2005 (UTC)
To fix redirects with a bot is easy. The hard problem is that not all the pages linking to derivation need to link to derivation (abstract algebra). In some cases the link should certainly be changed, in some cases, like the article dual numbers I am not sure. Could anybody knowing these things make a list of articles linking to derivation which should not be changed? Then I will run the bot on the ones which should be changed, which are a majority. Thanks. Oleg Alexandrov 20:17, 11 May 2005 (UTC)
I was very modestly thinking only of "all" the links on this page. MFH: Talk 21:48, 11 May 2005 (UTC)
Now I am really confused. There are just a couple of links to derivation (abstract algebra) on Lie derivative, and a couple on Talk:Lie derivative. I don't see any links to derivation. Also, if there were any, they would not be more than one or two, so can be fixed by hand. Am I misunderstanding something? Oleg Alexandrov 21:59, 11 May 2005 (UTC)

## Proposals/suggestions

I'm currently rewriting the mathematics of general relativity article and am writing stuff on the Lie derivative in that article. I have links to this (Lie derivative) article, but can I suggest that there should be more information here regarding the concept of Lie dragging a function along a congruence, as this leads nicely into the definition of a Lie derivative (perhaps as a section further in the article, not necessarily in the intro.). Another suggestion is to have component expressions for the Lie derivative of a tensor field and explain why (or at least state) that the Lie derivative does not need a connection for its definition (as is indicated by the fact that the covariant derivatives can be replaced by partials). Comments appreciated. Thanks. ---Mpatel (talk) 07:05, 29 September 2005 (UTC)

I don't have any books that mention Lie dragging, so I wouldn't know how to write that up. If its placed in its own section, that would be fine to me. This article already mentions three different definitions for a Lie derivative; having a fourth one is fine, as long as this definition is not presented as the first or primary definition.
The section on Lie derivatives of tensor fields is total garbage, and needs to be completely re-written. The connection should be banished. Somewhere in this article we should make clear that the connection is not required for any of this stuff; and explain why; since I think the interplay of a connection is a stumbling block for mastering these concepts. linas 14:06, 29 September 2005 (UTC)
Yep, it needs to go. I can recommend two possible approaches. The first is to define the Lie derivative as a derivation on the full tensor bundle of a differentiable manifold. This has the advantage of fitting into the overall algebraic tone of the article. The second is to define it by means of pullback under the one-parameter group of diffeomorphisms generated by a vector field: the vector flow. This has the advantage of saying what the Lie derivative actually means. Both approaches should have a place in the article. Silly rabbit 06:09, 8 November 2005 (UTC)
Now that I've read it, I think the whole article should be scrapped. Even the Lie derivative of a function is hopelessly confused! (For those of you readers who don't know what the Lie derivative of a function is, it's just the ordinary directional derivative. It's when you get to vector fields, differential forms, and tensor fields that things become interesting.) Silly rabbit 06:22, 8 November 2005 (UTC)
No reference whatsoever to the Picard-Lindelof or Frobenius theorem is required. Please see the definition of tangent space. It doesn't matter whether you get a particular flow through the point in question, only that there exists one with the required first order behavior. Those who know about such things (myself included) are aware of these theorems, but they add needless clutter to the article. Simply things as far as possible, and no further. Silly rabbit 06:45, 8 November 2005 (UTC)
Silly Rabbit, do note that the general consensus is that WP is a reference rather than a tutorial. To learn a new topic, a reader should read a book on the topic. In general, WP articles usually assume that the reader already has some general familiarity with the topic, possibly even a rather good understanding, but their knowledge is incomplete, or they are searching for something related that was once known, but forgotten (e.g. the Picard-Lindelof or Frobenius theorem) -- thus links and mentions like this are not "clutter" but are actually useful. Similarly, references to books or other online "tutorial" sources are important. There is a fairly strong feeling within WP that it is not a book or a tutorial, but a reference, although who knows what the future may bring. You are invited to debate issues of style and content at Wikipedia talk:WikiProject Mathematics. linas 14:50, 8 November 2005 (UTC)
I'm not sure that's right. The prejudice is against getting confused between WP and a textbook. Various tutorial techniques are perfectly OK. One person's helpful aside is another's buzzword generator, so I doubt that there is any absolute criterion of what to include. But the main thing is to aim at a comprehensive treatment of a topic. Well, that's in the Wikipedia charter; of course little in mathematics can actually be treated that thoroughly. Charles Matthews 15:45, 8 November 2005 (UTC)
Ok, I retract my anti-buzzword prejudice. But I still believe that the article could use some clarification and simplification. The algebraic methods are quite useful in practice, but they don't provide much insight into the meaning of the Lie derivative. Silly rabbit 02:07, 9 November 2005 (UTC)
Dear Linas, Alrighty then. I have extensively revised the article. In my opinion, it is now clearer, more complete, better organized, and contains fewer errors. I hope that my revisions satisfy you and the rest of the wikipedia math community. Best, Silly rabbit 06:10, 11 November 2005 (UTC) (P.S., I left the offending buzzwords intact, although I still believe that they ought to be removed or exported to an article like tangent bundle or vector field.)
I made some major revisions to Lie derivatives of tensor fields. There was some good stuff in there which I had to delete involving Lie derivatives of tensor densities. I have preserved it below in this talk page. Silly rabbit 03:00, 11 November 2005 (UTC)

A short question from a reader of this page: The definition of the Lie derivative given here makes it look like it eats vector fields (in both slots), but the function f that is introduced later on in the definition of the Lie derivative, and which the Lie derivative eats, is a function from M to R. Shouldn't this be a function from M to R^n, where n is the dimension of the manifold?

The definition is extended so that it eats also functions. A function from M to R^n would still be a function and NOT a tangent vector. --MarSch 13:17, 31 October 2005 (UTC)

Some suggestions about the list of axioms for Lie derivative of tensor fields.

• Axiom 3: I propose replacing axiom 3 with two more "primitive axioms":

and

Mungbean 13:17, 15 December 2005 (UTC)

### Removal or major edit of section "The Lie derivative of differential forms"

Second suggestion that we remove the definition and results concerning the interior product. These result are in the section interior product (Or should be) and should not be reproduced here. The section would now become:

The Lie derivative can also be defined on differential forms. In this context, it is closely related to the exterior derivative and the interior product.

The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. For an ordinary function f, the Lie derivative is just the contraction of the exterior derivative with the vector field X:

${\displaystyle {\mathcal {L}}_{X}f=i_{X}df}$

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:

${\displaystyle {\mathcal {L}}_{X}\omega =i_{X}d\omega +d(i_{X}\omega )}$.

The derivative of products is distributed:

${\displaystyle {\mathcal {L}}_{fX}\omega =f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega }$

Mungbean 12:36, 16 December 2005 (UTC)

Disagree. Note that different authors and different texts define the Lie derivative in different ways. Thus, all of these ways of approaching the definition should be described. No one particular definition should be elevated as "more right" than the others. A second important point is that many novices in the field find themselves rather confused by all the different derivatives, (Lie, exterior, partial and covariant) and have trouble sorting out the relationships between them. It is important for this article to succinctly state what these relationships are. That some of this info is also duplicated in other articles is acceptable. linas 21:33, 16 December 2005 (UTC)

## Lie derivatives of tensor densities

In differential geometry, if we have a differentiable tensor field ${\displaystyle T}$ of rank ${\displaystyle (p,q)}$ (which we consider as a differentiable linear map of smooth sections α, β, … of the cotangent bundle ${\displaystyle T^{*}M}$ and of sections X, Y, … of the tangent bundle ${\displaystyle TM}$, written T(α, β, …, X, Y, …), such that for any collection of smooth functions f1, …, fp, fp + 1, …, fp + q we have

${\displaystyle T(f_{1}\alpha ,f_{2}\beta ,\ldots ,f_{p+1}X,f_{p+2}Y,\ldots )=f_{1}f_{2}\cdots f_{p+1}f_{p+2}\cdots f_{p+q}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )}$,

and if further we have a differentiable vector field (i.e. a smooth section of the tangent bundle) ${\displaystyle A}$, then the linear map

${\displaystyle ({\mathcal {L}}_{A}T)(\alpha ,\beta ,\ldots ,X,Y,\ldots )\equiv \nabla _{A}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )-\nabla _{T(\cdot ,\beta ,\ldots ,X,Y,\ldots )}\alpha (A)-\ldots +T(\alpha ,\beta ,\ldots ,\nabla _{X}A,Y,\ldots )+\ldots }$

is independent of the connection ∇ used; as long as it is torsion-free, and in fact, is a tensor. This tensor is called the Lie derivative of ${\displaystyle T}$ with respect to ${\displaystyle A}$.

In other words, if you have a tensor field ${\displaystyle T}$ and an infinitesimal generator of a diffeomorphism given by a vector field ${\displaystyle U}$, then ${\displaystyle {\mathcal {L}}_{U}T}$ is nothing other than the infinitesimal change in ${\displaystyle T}$ under the infinitesimal diffeomorphism.

More generally, if we have a differentiable tensor density field ${\displaystyle T}$ of rank ${\displaystyle (p,q)}$ and density s, then

${\displaystyle ({\mathcal {L}}_{A}T)(\alpha ,\beta ,\ldots ,X,Y,\ldots )\equiv \nabla _{A}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )-\nabla _{T(\cdot ,\beta ,\ldots ,X,Y,\ldots )}\alpha (A)-\ldots +T(\alpha ,\beta ,\ldots ,\nabla _{X}A,Y,\ldots )+\ldots +s(\nabla \cdot A)T(\alpha ,\beta ,\ldots ,X,Y,\ldots )}$

Alternately, given the vector field ${\displaystyle U}$, let ψ be the family of integral curves of ${\displaystyle U}$, as given above. Note that ψ is a local 1-parameter group of local diffeomorphisms. Let ${\displaystyle \psi ^{*}}$ be the pullback induced by ψ. Then the Lie derivative of the tensor field ${\displaystyle T}$ at the point ${\displaystyle p}$ is given by

${\displaystyle {\mathcal {L}}_{U}T={\frac {d}{dt}}\left(\psi _{t}^{*}T\right)\vert _{\psi (t)=p}}$.

— Preceding unsigned comment added by Silly rabbit (talkcontribs) 03:03, 11 November 2005 (UTC)

## Definition is confusing

I see that several forms of the definition are mentioned. I think it would be less confusing to say something like: The definition in the case of tensor densities (or field) is the following ... In the following particular cases this boils down to simpler expressions: (i) Functions (that is, (0,0)-tensors):... (ii) Vector fields (that is (1,0)-tensors): ... And so on. 140.180.171.121 23:44, 23 September 2007 (UTC)

### for more than one reason

It is more confusing that it begins with "One might..." which suggests that a line of faulty reasoning is about to be dismissed (c.f. covariant derivative vs simpler derivative and the need to compensate for metric variation). Why not simply say "One can... in several ways..." but choose carefully and explain why each definition has particular value/insight.

Please review the updates to the definition of the Lie derivative for a function. I've removed all of the ugly "One" language. I haven't added any extra content providing some insight/value for each definition, but hopefully the more straight-forward presentations of each definition help. —TedPavlic (talk/contrib/@) 18:35, 22 July 2009 (UTC)

BTW - how is "Lie" pronounced? Lye or Lee? Julian I Do Stuff (talk) 15:36, 22 July 2009 (UTC)

Lee. I have no reference to confirm this to you though. —TedPavlic (talk/contrib/@) 18:35, 22 July 2009 (UTC)

## Unexplained equations

The explanation of the last two equations of section "The Lie derivative of a vector field" is completely missing. I talk about these equations:

${\displaystyle ({\mathcal {L}}_{X}Y)_{x}:=\lim _{t\to 0}(\mathrm {T} (\mathrm {Fl} _{-t}^{X})Y_{\mathrm {Fl} _{t}^{X}(x)}-Y_{x})/t=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}\mathrm {T} (\mathrm {Fl} _{-t}^{X})Y_{\mathrm {Fl} _{t}^{X}(x)}}$
${\displaystyle {\mathcal {L}}_{X}Y:=\left.{\frac {\mathrm {d} ^{2}}{2\mathrm {d} ^{2}t}}\right|_{t=0}\mathrm {Fl} _{-t}^{Y}\circ \mathrm {Fl} _{-t}^{X}\circ \mathrm {Fl} _{t}^{Y}\circ \mathrm {Fl} _{t}^{X}=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}\mathrm {Fl} _{-{\sqrt {t}}}^{Y}\circ \mathrm {Fl} _{-{\sqrt {t}}}^{X}\circ \mathrm {Fl} _{\sqrt {t}}^{Y}\circ \mathrm {Fl} _{\sqrt {t}}^{X}}$

It would be necessary to explain at least the meaning of ${\displaystyle \mathrm {Fl} \,}$. —Preceding unsigned comment added by 89.135.19.155 (talk) 06:17, 11 August 2008 (UTC)

I have never seen that notation before, so this is just a guess. "Fl" might be short for Flow (mathematics) which is a related concept. JRSpriggs (talk) 12:27, 11 August 2008 (UTC)
I'm familiar with the notation. "Fl" is indeed the flow, and T is the "tangent functor" or the pushforward (differential). siℓℓy rabbit (talk) 12:47, 11 August 2008 (UTC)
Yes it's clear that it "Fl" is the flow and T is the pushforward. The only problem that this isn't mentioned in the article.

89.135.19.155 (talk) 21:22, 11 August 2008 (UTC)

No not exactly the push forward, T is the Tangent functional (aka derivative) but the overall effect is push forward. Seems a pretty obscure way to write it to me.Billlion (talk) 21:16, 12 August 2008 (UTC)
This is an issue that I have been aware of for a long time, but have never gotten around to fixing. It should (and will... someday) be fixed. Thanks, siℓℓy rabbit (talk) 21:31, 11 August 2008 (UTC)

I am wondering what the original source of the equations on the second line is. The reference given at these equations (Kolar,Michor,Slovak) indeed provides a proof for it, but it doesn't give the origin of the equation either, nor does the second expression (with the square roots of t) appear in this book. — Preceding unsigned comment added by 94.224.97.175 (talk) 23:48, 1 December 2012 (UTC)

## Ambiguous notation

In the section "The Lie derivative of a function" it is not clear what is meant by the symbols ${\displaystyle X_{p}(f)\,}$ and ${\displaystyle (Xf)(p)\,}$. If it is simply meant that the defined entity (${\displaystyle ({\mathcal {L}}_{\!X}f)(p)}$) has different equivalent representations then instead of the symbol ${\displaystyle \triangleq }$ one should use ${\displaystyle \equiv }$. I would suggest that we change the definition of Lie derivative according to Khalil (see H.K.Khalil, Nonlinear Systems, Prentice Hall editions, 3rd edition, ISBN: 0-13-067389-7, London 2002, pp.509-501). So, let ${\displaystyle f,g:D\to \mathbb {R} }$ be sufficiently smooth functions and ${\displaystyle D\subseteq \mathbb {R} ^{n}}$. We define the Lie derivative of ${\displaystyle g}$ along ${\displaystyle f}$ as:

For me it's clear, see William M. Boothy "An introduction to differentiable manifolds and Riemannian Geometry", Academic Press, second edition, ISBN: 0-12-116052-1, Orlando 1986, p. 117, where one can read: "the function ${\displaystyle Xf\,}$, defined by ${\displaystyle (Xf)(p)=X_{p}f\,}$, is of class ${\displaystyle C^{r}\,}$" and p. 226, where one can find the formula: ${\displaystyle {\mathcal {L}}_{\!X}f=Xf}$. Your definition is "wrong". Remember, we are doing Differential Geometry. But I agree with you in that it is better to use the symbol ${\displaystyle \equiv \,}$, because ${\displaystyle Xf\,}$ is the result of a calculation. The Lie derivative of any object is always defined with respect a vector field ${\displaystyle X\,}$ (taking its flow), not with respect a function. Khalil's definition it's a particular one. It corresponds to taking the vector field ${\displaystyle X=f{\frac {\partial }{\partial x}}\,}$, in a particular chart and representation. Mgvongoeden (talk) 17:45, 3 November 2011 (UTC)

## Non-intuitive definition in lead for layman

The statement "The Lie derivative along a vector field is the evaluation of the vector field on functions" in the lead gives very little indication of how the vectors are to be considered, and in particular what the "evaluation" means. This assumes context that is simply missing from the description, namely that the vectors are to be treated as operators, specifically directional derivatives. The subsequent statement "the Lie derivative is the commutator" does not clarify this for the non-initiate, since the commutator does not define what product is involved; in this case it is evidently the operator composition. This can be especially confusing to someone who is thinking of vectors simply in terms of a basis, and is expecting, for example, tensor products, but is faced with an undefined juxtaposition in the article commutator. Juxtaposition of two vectors can, for example, be interpreted as the geometric product.

I'm sure it would only take a few words to mention the interpretation of vectors (or at least their evaluation over a scalar field) as operators (specifically directional derivatives) and that the commutator relates to their use and composition as operators (functionals). This is not my field, and I request someone more familiar with it to clarify this. — Quondum 11:32, 24 June 2012 (UTC)

## Mumbo jumbo...

Well, for somebody who is used to find clarity searching on wikipedia, this article is clearly written in mumbo jumbo language. So I've checked also the Wolfram mathworld for some clue, any clue...and there really is one: http://mathworld.wolfram.com/LieDerivative.html . Also this wiki article lacks the importance and significance of this derivative in physics. So clearly it is just another strange expression in Mumbo jumbo language...Theodore Yoda (talk) 20:15, 29 January 2013 (UTC)