Talk:Lie derivative

message: 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)
replies: text:  Added a motivation paragraph to the beginning of the article David9550 (talk) 17:26, 5 June 2016 (UTC)
thread_title:  Motivation needed 

message: (sorry for my very bad English)
-  	: [A,B] := \mathcal{L}_A B = - \mathcal{L}_B A   	
+  	: [A,B] := \mathcal{L}_A B - \mathcal{L}_B A
The first equation is correct.  The second is not. 69.129.194.27 05:23, 22 July 2006 (UTC)
thread_title:  Where is the sign = ? where is the true ? 

message: 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 \mathrm{vol} for the volume form. I think it is less confusing. -MarSch 15:56, 27 Apr 2005 (UTC)
replies: text: 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)
replies: text: 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)
thread_title:  omega 

message: Has somebody a bot who could fix all links to derivation by adding (abstract algebra)? — MFH: Talk 22:20, 10 May 2005 (UTC)
replies: text: ask User:Oleg Alexandrov, he has a bot. -MarSch 14:43, 11 May 2005 (UTC)
replies: text:  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)
replies: text:  I was very modestly thinking only of "all" the links on this page. — MFH: Talk 21:48, 11 May 2005 (UTC)
replies: text:  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)
thread_title:  Fix links to derivation 

message: 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:
:\mathcal{L}_Xf = i_X df
For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:
:\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega).
The derivative of products is distributed:
:\mathcal{L}_{fX}\omega = 
f\mathcal{L}_X\omega + df \wedge i_X \omega
Mungbean 12:36, 16 December 2005 (UTC)
replies: text: 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)
thread_title: Removal or major edit of section "The Lie derivative of differential forms"

message: In differential geometry, if we have a differentiable tensor field T of rank (p,q) (which we consider as a differentiable linear map of smooth sections α, β, … of the cotangent bundle T^*M and of sections X, Y, … of the tangent bundle TM, written T(α, β, …, X, Y, …), such that for any collection of smooth functions f1, …, fp, fp + 1, …, fp + q we have
: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) A, then the linear map 
:(\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 T with respect to A.
In other words, if you have a tensor field T and an infinitesimal generator of a diffeomorphism given by a vector field U, then \mathcal{L}_{U} T is nothing other than the infinitesimal change in T under the infinitesimal diffeomorphism.
More generally, if we have a differentiable tensor density field T of rank (p,q) and density s, then
:(\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)
where \nabla\cdot A is the divergence.
Alternately, given the vector field U, let ψ be the family of integral curves of U, as given above.  Note that ψ is  a local 1-parameter group of local diffeomorphisms.  Let \psi^* be the pullback induced by ψ. Then the Lie derivative of the tensor field T at the point p is given by
:\mathcal{L}_U T = \frac{d}{dt}\left(\psi^*_t T\right) \vert_{\psi(t)=p}.
— Preceding unsigned comment added by  Silly rabbit  (talk • contribs)  03:03, 11 November 2005 (UTC)
thread_title:  Lie derivatives of tensor densities 

message: 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)
replies: text:  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)
replies: text:  Lee. I have no reference to confirm this to you though. —TedPavlic (talk/contrib/@) 18:35, 22 July 2009 (UTC)
thread_title: Definition is confusing

message: 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)
replies: text:  Lee. I have no reference to confirm this to you though. —TedPavlic (talk/contrib/@) 18:35, 22 July 2009 (UTC)
thread_title: for more than one reason

message: The explanation of the last two equations of section "The Lie derivative of a vector field" is completely missing.
I talk about these equations:
:(\mathcal{L}_X Y)_x := \lim_{t \to 0} (\mathrm{T}(\mathrm{Fl}^X_{-t}) Y_{\mathrm{Fl}^X_t(x)} - Y_x)/t = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} \mathrm{T}(\mathrm{Fl}^X_{-t}) Y_{\mathrm{Fl}^X_t(x)}
:\mathcal{L}_X Y := \left.\frac{\mathrm{d}^2}{2\mathrm{d}^2 t}\right|_{t=0} \mathrm{Fl}^Y_{-t} \circ \mathrm{Fl}^X_{-t} \circ \mathrm{Fl}^Y_{t} \circ \mathrm{Fl}^X_{t} = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} \mathrm{Fl}^Y_{-\sqrt{t}} \circ \mathrm{Fl}^X_{-\sqrt{t}} \circ \mathrm{Fl}^Y_{\sqrt{t}} \circ \mathrm{Fl}^X_{\sqrt{t}}
It would be necessary to explain at least the meaning of \mathrm{Fl}\,.  —Preceding unsigned comment added by 89.135.19.155 (talk) 06:17, 11 August 2008 (UTC)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title:  Unexplained equations 

message: In the section "The Lie derivative of a function" it is not clear what is meant by the symbols X_p(f)\, and (Xf)(p)\,. If it is simply meant that the defined entity ((\mathcal{L}_{\!X} f)(p)) has different equivalent representations then instead of the symbol \triangleq one should use \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 f,g:D\to\mathbb{R} be sufficiently smooth functions and D\subseteq\mathbb{R}^n. We define the Lie derivative of g along f as:
(L_f g)(x)\triangleq(\frac{\partial g}{\partial x}f)(x)
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 Xf\,,  defined by (Xf)(p)=X_pf\,, is of class C^r\," and p. 226, where one can find the formula: \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 \equiv\,, because Xf\, is the result of a calculation. The Lie derivative of any object is always defined with respect a vector field X\, (taking its flow), not with respect a function. Khalil's definition it's a particular one. It corresponds to taking the vector field X=f\frac{\partial }{\partial x}\,, in a particular chart and representation. Mgvongoeden (talk) 17:45, 3 November 2011 (UTC)
thread_title:  Ambiguous notation 

message: 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)
thread_title:  Non-intuitive definition in lead for layman 

message: 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)
thread_title:  Mumbo jumbo... 

message: A few days ago I added a new subsection with some examples. I think they are very useful, especially for people who are now learning about the Lie derivative and since the article is definitely not one of the most clear ones on Wikipedia. User:Mgvongoeden removed them without any discussion or explanation whatsoever though, so I reinstated them. Other articles like the one for the covariant derivative have a small subsection with examples so it's obvious that having the examples really helps people understand the material.129.194.8.73 (talk) 10:03, 16 January 2015 (UTC)
thread_title:  Examples 

message: I have no explanation for the undo of the following lines. I will add them again. If somebody wants to erase them, first we should have  a discussion. This notation exists and I personally believe it is much better. A good reference to find this notation is the following book (see introduction): http://matematicas.unex.es/~navarro/degree.pdf
"Another notation to denote the Lie derivative  \mathcal{L}_X(Y) , although less common, is  X^L Y . This notation is usually cleaner since it avoids the use of subindexes. This notation comes from prof. Juan Bautista Sancho Guimerá"". JBecerra (talk) 13:49, 16 January 2018 (UTC)
thread_title:  Sancho Guimerá's notation 

message: Is there any source that uses this axiom to define Lie derivative?  If not, what is the meaning of the expression \mathcal{L}_X(T(Y_1,\ldots,T_n)) on the left-hand side? StrokeOfMidnight (talk) 03:54, 25 November 2020 (UTC)
thread_title:  The Lie derivative of a tensor field: axiom 3