Editing Lie derivative

{{Short description|A derivative in Differential Geometry}}
{{Use dmy dates|date=November 2024}}
In [[differential geometry]], the '''Lie derivative''' ({{IPAc-en|l|iː}} {{respell|LEE}}), named after [[Sophus Lie]] by [[Władysław Ślebodziński]],<ref>{{cite book |first=A. |last=Trautman |author-link=Andrzej Trautman |year=2008 |chapter=Remarks on the history of the notion of Lie differentiation |title=Variations, Geometry and Physics: In honour of Demeter Krupka's sixty-fifth birthday |editor1-first=O. |editor1-last=Krupková |editor2-first=D. J. |editor2-last=Saunders |location=New York |publisher=Nova Science |isbn=978-1-60456-920-9 |pages=297–302 }}</ref><ref>{{cite journal |last=Ślebodziński |first=W. |year=1931 |title=Sur les équations de Hamilton |journal=Bull. Acad. Roy. D. Belg. |volume=17 |issue=5 |pages=864–870 }}</ref> evaluates the change of a [[tensor field]] (including scalar functions, [[vector field]]s and [[one-form]]s), along the [[flow (mathematics)|flow]] defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any  [[differentiable manifold]].

Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted <math> \mathcal{L}_X T</math>. The [[differential operator]] <math> T \mapsto \mathcal{L}_X T</math> is a [[derivation (differential algebra)|derivation]] of the algebra of [[tensor fields]] of the underlying manifold.

The Lie derivative commutes with [[Tensor contraction|contraction]] and the [[exterior derivative]] on [[differential forms]].

Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or [[scalar field]]. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.

The Lie derivative of a vector field ''Y'' with respect to another vector field ''X'' is known as the "[[Lie bracket of vector fields|Lie bracket]]"  of ''X'' and ''Y'', and is often denoted [''X'',''Y''] instead of <math> \mathcal{L}_X Y</math>. The space of vector fields forms a [[Lie algebra]] with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional [[Lie algebra representation]] of this Lie algebra, due to the identity

:<math> \mathcal{L}_{[X,Y]} T = \mathcal{L}_X \mathcal{L}_{Y} T - \mathcal{L}_Y \mathcal{L}_X T,</math>

{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
| <strong>Proof of the identity</strong>
|-
| :<math> \mathcal{L}_{[X,Y]} T = [[X,Y],T] = [X,Y]T - T[X,Y] = ([X,YT] - Y[X,T]) - ([X,TY] - [X,T]Y) = </math>
:<math> = XYT - YTX - Y[X,T] - XTY + TYX + [X,T]Y = (XYT - YTX) + (TYX - XTY) - (Y[X,T] - [X,T]Y) =</math>
:<math> = (X[Y,T] + [T,Y]X) - (Y[X,T] - [X,T]Y) = (X[Y,T] - [Y,T]X) - (Y[X,T] - [X,T]Y) = </math>
:<math> = [X,[Y,T]] - [Y,[X,T]] = \mathcal{L}_X \mathcal{L}_{Y} T - \mathcal{L}_Y \mathcal{L}_X T</math>
|}

valid for any vector fields ''X'' and ''Y'' and any tensor field ''T''.

Considering vector fields as [[Lie algebra|infinitesimal generator]]s of [[Flow (mathematics)|flows]] (i.e. one-dimensional [[Group (mathematics)|groups]] of [[diffeomorphism]]s) on ''M'', the Lie derivative is the [[Lie algebra representation#Infinitesimal Lie group representations|differential]] of the representation of the [[Diffeomorphism#Diffeomorphism group|diffeomorphism group]] on tensor fields, analogous to Lie algebra representations as [[Lie algebra representation#Infinitesimal Lie group representations|infinitesimal representations]] associated to [[group representation]] in [[Lie group]] theory.

Generalisations exist for [[spinor]] fields, [[fibre bundle]]s with a [[Connection (mathematics)|connection]] and [[vector-valued differential forms]].

==Motivation==
A 'naïve' attempt to define the derivative of a [[tensor field]] with respect to a [[vector field]] would be to take the [[Tensor#As multidimensional arrays|components]] of the tensor field and take the [[directional derivative]] of each component with respect to the vector field. However, this definition is undesirable because it is not invariant under [[Manifold#Transition map|changes of coordinate system]], e.g. the naive derivative expressed in [[polar coordinate system|polar]] or [[spherical coordinate system|spherical coordinates]] differs from the naive derivative of the components in [[Cartesian coordinate system|Cartesian coordinates]]. On an abstract [[manifold]] such a definition is meaningless and ill defined.

In [[differential geometry]], there are three main coordinate independent notions of differentiation of tensor fields:

# Lie derivatives,
# derivatives with respect to [[connection (differential geometry)|connections]],
# the [[exterior derivative]] of totally antisymmetric covariant tensors, i.e. [[differential forms]].

The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a [[tangent space|tangent vector]] is well-defined even if it is not specified how to extend that tangent vector to a vector field. However, a connection requires the choice of an additional geometric structure (e.g. a [[Riemannian manifold|Riemannian metric]] in the case of [[Levi-Civita connection]], or just an abstract [[connection (differential geometry)|connection]]) on the manifold. In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field ''X'' at a point ''p'' depends on the value of ''X'' in a neighborhood of ''p'', not just at ''p'' itself. Finally, the exterior derivative of differential forms does not require any additional choices, but is only a well defined derivative of differential forms (including functions), thus excluding vectors and other tensors that are not purely differential forms.
[[File:Lie transport.jpg|thumb|Lie transport of a vector <math>v_y</math> from point <math>y</math> to point <math>x</math> along the vector flow field <math>u</math>.]]
The idea of Lie derivatives is to use a vector field to define a notion of transport (Lie transport). A smooth vector field defines a smooth flow on the manifold, which allows vectors to be transported between two points on the same line of flow (This contrasts with connections, which allows transport between arbitrary points). Intuitively, a vector <math>Y(p)</math> based at point <math>p</math> is transported by flowing its base point to <math>p'</math>, while flowing its tip point <math>p + Y(p) \delta</math> to <math>p' + \delta p'</math>.

==Definition==
The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.

===The (Lie) derivative of a function===
Defining the derivative of a function <math>f\colon M \to {\mathbb R} </math> on a manifold takes care because the [[difference quotient]] <math>\textstyle (f(x+h)-f(x))/h </math> cannot be determined while the displacement <math>x+h</math> is undefined.

The Lie derivative of a function <math>f\colon M\to {\mathbb R}</math> with respect to a [[vector field]] <math>X</math> at a point <math>p \in M</math> is the function
:<math>(\mathcal{L}_X f) (p) = {d \over dt} \biggr|_{t=0} \bigl(f \circ \Phi^t_X\bigr)(p) = \lim_{t\to 0} \frac{f\bigl(\Phi^t_X(p)\bigr) - f\bigl(p\bigr)}{t}</math>
where <math>\Phi^t_X(p)</math> is the point to which the [[flow (mathematics)|flow]] defined by the vector field <math>X</math> maps the point <math>p</math> at time instant <math>t.</math> In the vicinity of <math>t=0,</math> <math>\Phi^t_X(p)</math> is the unique solution of the system
:<math>
\frac{d}{dt}\biggr|_t \Phi^t_X(p) = X\bigl(\Phi^t_X(p)\bigr)
</math>
of first-order autonomous (i.e. time-independent) differential equations, with <math>\Phi^0_X(p) = p.</math>

Setting <math>\mathcal{L}_X f = \nabla_X f</math> identifies the Lie derivative of a function with the [[directional derivative]], which is also denoted by <math> X(f):= \mathcal{L}_X f = \nabla_X f</math>.

===The Lie derivative of a vector field===
If ''X'' and ''Y'' are both vector fields, then the Lie derivative of ''Y'' with respect to ''X'' is also known as the [[Lie bracket of vector fields|Lie bracket]] of ''X'' and ''Y'', and is sometimes denoted <math>[X,Y]</math>. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:

{{unordered list
| The Lie bracket of ''X'' and ''Y'' at ''p'' is given in local coordinates by the formula
: <math>\mathcal{L}_X Y (p) = [X,Y](p) = \partial_X Y(p) - \partial_Y X(p),</math>

where <math>\partial_X</math> and <math>\partial_Y</math> denote the operations of taking the [[directional derivative]]s with respect to ''X'' and ''Y'', respectively. Here we are treating a vector in ''n''-dimensional space as an ''n''-[[tuple]], so that its directional derivative is simply the tuple consisting of the directional derivatives of its coordinates. Although the final expression <math>\partial_X Y(p) - \partial_Y X(p)</math> appearing in this definition does not depend on the choice of local coordinates, the individual terms <math>\partial_X Y(p)</math> and <math>\partial_Y X(p)</math> do depend on the choice of coordinates.
| If ''X'' and ''Y'' are vector fields on a manifold ''M'' according to the second definition, then the operator <math>\mathcal{L}_X Y = [X,Y]</math> defined by the formula
: <math>[X,Y]: C^\infty(M) \rightarrow C^\infty(M)</math>
: <math>[X,Y](f) = X(Y(f)) - Y(X(f))</math>

is a derivation of order zero of the algebra of smooth functions of ''M'', i.e. this operator is a vector field according to the second definition.
}}

===The Lie derivative of a tensor field===
====Definition in terms of flows====
The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow.

Formally, given a differentiable (time-independent) vector field <math>X</math> on a smooth manifold <math>M,</math> let <math>\Phi^t_X : M \to M</math> be the corresponding local flow. Since <math>\Phi^t_X</math> is a local diffeomorphism for each <math>t</math>, it gives rise to a [[Pullback (differential geometry)#Pullback by diffeomorphisms|pullback of tensor fields]]. For covariant tensors, this is just the multi-linear extension of the  [[pullback (differential geometry)|pullback map]]

<math display=block>\left(\Phi^t_X\right)^*_p : T^*_{\Phi^t_X(p)}M \to T^*_{p}M, \qquad \left(\left(\Phi^t_X\right)^*_p \alpha\right) (Y) = \alpha\bigl(T_p \Phi^t_X(Y)\bigr), \quad \alpha \in T^*_{\Phi^t_X(p)}M, Y \in T_{p}M  </math>
For contravariant tensors, one extends the inverse
:<math>\left(T_p\Phi^t_X\right)^{-1} : T_{\Phi^t_X(p)}M \to T_{p}M</math>

of the [[pushforward (differential)|differential]] <math>T_p\Phi^t_X </math>. For every <math>t,</math> there is, consequently, a tensor field <math>(\Phi^t_X)^* T</math> of the same type as <math>T</math>'s.

If <math>T</math> is an <math>(r,0)</math>- or <math>(0,s)</math>-type tensor field, then the Lie derivative <math>{\cal L}_XT</math> of <math>T</math> along a vector field <math>X</math> is defined at point <math>p \in M</math> to be

:<math>{\cal L}_X T(p) = \frac{d}{dt}\biggl|_{t=0} \left(\bigl(\Phi^t_X\bigr)^* T\right)_p = \frac{d}{dt}\biggl|_{t=0}\bigl(\Phi^t_X\bigr)^*_p T_{\Phi^t_X(p)}
= \lim_{t \to 0}\frac{\bigl(\Phi^t_X\bigr)^*T_{\Phi^t_X(p)} - T_p}{t}.</math>

The resulting tensor field <math>{\cal L}_X T</math> is of the same type as <math>T</math>'s.



More generally, for every smooth 1-parameter family <math>\Phi_t</math> of diffeomorphisms that integrate a vector field <math>X </math> in the sense that <math>{d \over dt}\biggr|_{t=0} \Phi_t = X \circ \Phi_0 </math>, one has<math display="block">\mathcal{L}_X T = \bigl(\Phi_0^{-1}\bigr)^* {d \over dt}\biggr|_{t=0} \Phi_t^* T = - {d \over dt}\biggr|_{t=0} \bigl(\Phi_t^{-1}\bigr)^* \Phi_0^* T \, . </math>

====Algebraic definition====
We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms:

:'''Axiom 1.''' The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula
::<math>\mathcal{L}_Yf=Y(f)</math>

:'''Axiom 2.''' The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields ''S'' and ''T'', we have
::<math>\mathcal{L}_Y(S\otimes T)=(\mathcal{L}_YS)\otimes T+S\otimes (\mathcal{L}_YT).</math>

:'''Axiom 3.''' The Lie derivative obeys the Leibniz rule with respect to [[Tensor contraction|contraction]]:
::<math> \mathcal{L}_X (T(Y_1, \ldots, Y_n)) = (\mathcal{L}_X T)(Y_1,\ldots, Y_n) + T((\mathcal{L}_X Y_1), \ldots, Y_n) + \cdots + T(Y_1, \ldots, (\mathcal{L}_X Y_n)) </math>

:'''Axiom 4.''' The Lie derivative commutes with exterior derivative on functions:
::<math> [\mathcal{L}_X, d] = 0 </math>

Using the first and third axioms, applying the Lie derivative <math>\mathcal{L}_X</math> to <math> Y(f) </math> shows that
::<math>\mathcal{L}_X Y (f) = X(Y(f)) - Y(X(f)),</math>
which is one of the standard definitions for the [[Lie bracket of vector fields|Lie bracket]].

The Lie derivative acting on a differential form is the [[Commutator#Ring theory|anticommutator]] of the [[interior product]] with the exterior derivative. So if α is a differential form,
::<math>\mathcal{L}_Y\alpha=i_Yd\alpha+di_Y\alpha.</math>
This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions. This is '''Cartan's magic formula'''. See [[interior product]] for details.

Explicitly, let ''T'' be a tensor field of type {{nowrap|(''p'', ''q'')}}. Consider ''T'' to be a differentiable [[multilinear map]] of [[smooth function|smooth]] [[section (fiber bundle)|sections]] ''α''<sup>1</sup>, ''α''<sup>2</sup>, ..., ''α''<sup>''p''</sup> of the cotangent bundle ''T''<sup>∗</sup>''M'' and of sections ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>q</sub> of the [[tangent bundle]] ''TM'', written ''T''(''α''<sup>1</sup>, ''α''<sup>2</sup>, ..., ''X''<sub>1</sub>, ''X''<sub>2</sub>, ...) into '''R'''. Define the Lie derivative of ''T'' along ''Y'' by the formula

:<math>(\mathcal{L}_Y T)(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) =Y(T(\alpha_1,\alpha_2,\ldots,X_1,X_2,\ldots))</math>
::<math>- T(\mathcal{L}_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots)
- T(\alpha_1, \mathcal{L}_Y\alpha_2, \ldots, X_1, X_2, \ldots) -\ldots </math>
::<math>- T(\alpha_1, \alpha_2, \ldots, \mathcal{L}_YX_1, X_2, \ldots)
-  T(\alpha_1, \alpha_2, \ldots, X_1, \mathcal{L}_YX_2, \ldots) - \ldots
</math>

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the [[General Leibniz rule|Leibniz rule]] for differentiation. The Lie derivative commutes with the contraction.

===The Lie derivative of a differential form===
{{see also|Interior product}}
A particularly important class of tensor fields is the class of [[differential forms]]. The restriction of the Lie derivative to the space of differential forms is closely related to the [[exterior derivative]]. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an [[interior product]], after which the relationships falls out as an identity known as '''Cartan's formula'''. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.

Let ''M'' be a manifold and ''X'' a vector field on ''M''. Let <math>\omega \in \Lambda^k(M)</math> be a ''k''-[[Differential form|form]], i.e., for each <math>p \in M</math>, <math>\omega(p)</math> is an [[Alternating form|alternating]] [[multilinear map]] from <math>(T_p M)^k</math> to the real numbers. The [[interior product]] of ''X'' and ''ω'' is the {{nowrap|(''k'' &minus; 1)}}-form <math>i_X\omega</math> defined as

:<math>(i_X\omega) (X_1, \ldots, X_{k-1}) = \omega (X,X_1, \ldots, X_{k-1})\,</math>

The differential form <math>i_X\omega</math> is also called the '''contraction''' of ''ω'' with ''X'', and
:<math>i_X:\Lambda^k(M) \rightarrow \Lambda^{k-1}(M)</math>

is a [[Exterior algebra | <math>\wedge</math>]]-[[derivation (abstract algebra)|antiderivation]] where [[Exterior algebra |<math>\wedge</math>]] is the [[Exterior algebra |wedge product on differential forms]]. That is, <math>i_X</math> is '''R'''-linear, and

:<math>i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta)</math>

for <math>\omega \in \Lambda^k(M)</math> and η another differential form. Also, for a function <math>f \in \Lambda^0(M)</math>, that is, a real- or complex-valued function on ''M'', one has

:<math>i_{fX} \omega = f\,i_X\omega</math>

where <math>f X</math> denotes the product of ''f'' and ''X''.
The relationship between [[exterior derivative]]s and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function ''f'' with respect to a vector field ''X'' is the same as the directional derivative ''X''(''f''), it is also the same as the [[Differential form#Operations on forms|contraction]] of the exterior derivative of ''f'' with ''X'':

:<math>\mathcal{L}_Xf = i_X \, df</math>

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

:<math>\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega).</math>

This identity is known variously as '''Cartan formula''', '''Cartan homotopy formula''' or '''Cartan's magic formula'''. See [[interior product]] for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that

:<math>d\mathcal{L}_X\omega = \mathcal{L}_X(d\omega).</math>

The Lie derivative also satisfies the relation

:<math>\mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega .</math>

==Coordinate expressions==
{{Einstein summation convention}}

In local [[coordinate]] notation, for a type {{nowrap|(''r'', ''s'')}} tensor field <math>T</math>, the Lie derivative along <math>X</math> is
:<math>\begin{align}
  (\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} ={}
    & X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) \\
    & {}-{} (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} \\
    & + (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c}
\end{align}</math>

here, the notation <math>\partial_a = \frac{\partial}{\partial x^a}</math> means taking the partial derivative with respect to the coordinate <math>x^a</math>. Alternatively, if we are using a [[torsion (differential geometry)|torsion-free]] [[connection (mathematics)|connection]] (e.g., the [[Levi Civita connection]]), then the partial derivative <math>\partial_a</math> can be replaced with the [[covariant derivative]] which means replacing <math>\partial_a X^b</math> with (by abuse of notation) <math>\nabla_a X^b = X^b_{;a} := (\nabla X)_a^{\ b} = \partial_a X^b + \Gamma^b_{ac}X^c</math> where the <math>\Gamma^a_{bc} = \Gamma^a_{cb}</math> are the [[Christoffel coefficients]].

The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor
:<math>(\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}\partial_{a_1}\otimes\cdots\otimes\partial_{a_r}\otimes dx^{b_1}\otimes\cdots\otimes dx^{b_s}</math>
which is independent of any coordinate system and of the same type as <math>T</math>.

The definition can be extended further to [[tensor densities]].  If ''T'' is a tensor density of some real number valued weight ''w'' (e.g. the volume density of weight 1), then its Lie derivative is a tensor density of the same type and weight.
:<math>\begin{align}
  (\mathcal {L}_X T)^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} ={}
    &X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) - (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} + \\
    &+ (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} + w (\partial_{c} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s}}
\end{align}</math>

Notice the new term at the end of the expression.

For a [[Affine connection|linear connection]] <math>\Gamma = ( \Gamma^{a}_{bc} )</math>, the Lie derivative along <math>X</math> is<ref>{{cite book|author-link=Kentaro Yano (mathematician) |last=Yano |first=K. |title=The Theory of Lie Derivatives and its Applications
|url=https://archive.org/details/theoryofliederiv029601mbp|publisher=North-Holland|year=1957|page=[https://archive.org/details/theoryofliederiv029601mbp/page/n25 8]|isbn=978-0-7204-2104-0}}</ref>
:<math>
(\mathcal{L}_X \Gamma)^{a}_{bc} =  X^d\partial_d \Gamma^{a}_{bc} + \partial_b\partial_c X^a - \Gamma^{d}_{bc}\partial_d X^a + \Gamma^{a}_{dc}\partial_b X^d + \Gamma^{a}_{bd}\partial_c X^d</math>

===Examples===
For clarity we now show the following examples in local [[coordinate]] notation.

For a [[scalar field]] <math>\phi(x^c)\in\mathcal{F}(M)</math> we have:
:<math> (\mathcal {L}_X \phi) = X(\phi) = X^a \partial_a \phi</math>.

Hence for the scalar field <math>\phi(x,y) = x^2 - \sin(y)</math> and the vector field <math>X^a\partial_a = \sin(x)\partial_y - y^2\partial_x</math> the corresponding Lie derivative becomes 
<math display="block">\begin{alignat}{3}
\mathcal{L}_X\phi &= (\sin(x)\partial_y - y^2\partial_x)(x^2 - \sin(y))\\
                  & = \sin(x)\partial_y(x^2 - \sin(y)) - y^2\partial_x(x^2 - \sin(y))\\
                  & = -\sin(x)\cos(y) - 2xy^2 \\
\end{alignat}</math>

For an example of higher rank differential form, consider the 2-form <math>\omega = (x^2 + y^2)dx\wedge dz</math> and the vector field <math>X</math> from the previous example. Then,
<math display="block">\begin{align}
\mathcal{L}_X\omega & = d(i_{\sin(x)\partial_y - y^2\partial_x}((x^2 + y^2)dx\wedge dz)) + i_{\sin(x)\partial_y - y^2\partial_x}(d((x^2 + y^2)dx\wedge dz)) \\
& = d(-y^2(x^2 + y^2) dz) + i_{\sin(x)\partial_y - y^2\partial_x}(2ydy\wedge dx\wedge dz) \\
& = \left(- 2xy^2 dx + (-2yx^2  - 4y^3) dy\right) \wedge dz + (2y\sin(x)dx \wedge dz + 2y^3dy \wedge dz)\\ 
& = \left(-2xy^2 + 2y\sin(x)\right)dx\wedge dz + (-2yx^2 - 2y^3)dy\wedge dz
\end{align}</math>

Some more abstract examples.
:<math>\mathcal{L}_X (dx^b) = d i_X (dx^b) = d X^b = \partial_a X^b dx^a </math>.

Hence for a [[one-form|covector field]], i.e., a [[differential form]], <math>A = A_a(x^b)dx^a</math> we have:
:<math>\mathcal{L}_X A =  X (A_a) dx^a +  A_b \mathcal{L}_X (dx^b) = (X^b \partial_b A_a + A_b\partial_a (X^b))dx^a</math>

The coefficient of the last expression is the local coordinate expression of the Lie derivative. 

For a covariant rank 2 tensor field <math>T = T_{ab}(x^c)dx^a \otimes dx^b</math> we have:
<math display="block">\begin{align} 
(\mathcal {L}_X T) &= (\mathcal {L}_X T)_{ab} dx^a\otimes dx^b\\
                   &= X(T_{ab})dx^a\otimes dx^b + T_{cb} \mathcal{L}_X (dx^c) \otimes dx^b + T_{ac}  dx^a \otimes \mathcal{L}_X (dx^c)\\  
                   &= (X^c \partial_c T_{ab}+T_{cb}\partial_a X^c+T_{ac}\partial_b X^c)dx^a\otimes dx^b\\
\end{align}</math>

If <math>T = g</math> is the symmetric metric tensor, it is parallel with respect to the [[Levi-Civita connection]] (aka [[covariant derivative]]), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving
:<math>(\mathcal {L}_X g) = (X^c g_{ab; c} + g_{cb}X^c_{;a} + g_{ac}X^c_{; b})dx^a\otimes dx^b = (X_{b;a} + X_{a;b}) dx^a\otimes dx^b</math>

==Properties==
The Lie derivative has a number of properties. Let <math>\mathcal{F}(M)</math> be the [[algebra over a field|algebra]] of functions defined on the [[manifold]] ''M''. Then

:<math>\mathcal{L}_X : \mathcal{F}(M) \rightarrow \mathcal{F}(M)</math>

is a [[derivation (abstract algebra)|derivation]] on the algebra <math>\mathcal{F}(M)</math>. That is,
<math>\mathcal{L}_X</math> is '''R'''-linear and

:<math>\mathcal{L}_X(fg) = (\mathcal{L}_Xf) g + f\mathcal{L}_Xg.</math>

Similarly, it is a derivation on <math>\mathcal{F}(M) \times \mathcal{X}(M)</math> where <math>\mathcal{X}(M)</math> is the set of vector fields on ''M'':<ref>{{Cite journal |last=Nichita |first=Florin F. |date=2019 |title=Unification Theories: New Results and Examples |journal=Axioms |language=en |volume=8 |issue=2 |at=p.60, Theorem 6 |doi=10.3390/axioms8020060 |doi-access=free |issn=2075-1680}}</ref>

:<math>\mathcal{L}_X(fY) = (\mathcal{L}_Xf) Y + f\mathcal{L}_X Y</math>

which may also be written in the equivalent notation

:<math>\mathcal{L}_X(f\otimes Y) = (\mathcal{L}_Xf) \otimes Y + f\otimes \mathcal{L}_X Y</math>

where the [[tensor product]] symbol <math>\otimes</math> is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.

Additional properties are consistent with that of the [[Lie bracket of vector fields|Lie bracket]]. Thus, for example, considered as a derivation on a vector field,

:<math>\mathcal{L}_X [Y,Z] = [\mathcal{L}_X Y,Z] + [Y,\mathcal{L}_X Z]</math>

one finds the above to be just the [[Jacobi identity]]. Thus, one has the important result that the space of vector fields over ''M'', equipped with the Lie bracket, forms a [[Lie algebra]].

The Lie derivative also has important properties when acting on differential forms. Let ''α'' and ''β'' be two differential forms on ''M'', and let ''X'' and ''Y'' be two vector fields. Then
* <math>\mathcal{L}_X(\alpha\wedge\beta) = (\mathcal{L}_X\alpha) \wedge\beta + \alpha\wedge (\mathcal{L}_X\beta)</math>
* <math>[\mathcal{L}_X,\mathcal{L}_Y]\alpha := \mathcal{L}_X\mathcal{L}_Y\alpha-\mathcal{L}_Y\mathcal{L}_X\alpha = \mathcal{L}_{[X,Y]}\alpha</math>
* <math>[\mathcal{L}_X,i_Y]\alpha = [i_X,\mathcal{L}_Y]\alpha = i_{[X,Y]}\alpha,</math> where ''i'' denotes interior product defined above and it is clear whether [·,·] denotes the [[commutator]] or the [[Lie bracket of vector fields]].

==Generalizations==
Various generalizations of the Lie derivative play an important role in differential geometry.

===The Lie derivative of a spinor field===
A definition for Lie derivatives of [[spinors]] along generic spacetime vector fields, not necessarily [[Killing vector field|Killing]] ones, on a general (pseudo) [[Riemannian manifold]] was already proposed in 1971 by [[Yvette Kosmann-Schwarzbach|Yvette Kosmann]].<ref name="autogenerated317">{{cite journal |last=Kosmann |first=Y. |author-link=Yvette Kosmann-Schwarzbach |year=1971 |title=Dérivées de Lie des spineurs |journal=[[Annali di Matematica Pura ed Applicata|Ann. Mat. Pura Appl.]] |volume=91 |issue=4 |pages=317–395 |doi=10.1007/BF02428822 |s2cid=121026516 }}</ref> Later, it was provided a geometric framework which justifies her ''ad hoc'' prescription within the general framework of Lie derivatives on [[fiber bundles]]<ref>{{cite book |last=Trautman |first=A. |year=1972 |chapter=Invariance of Lagrangian Systems |editor-first=L. |editor-last=O'Raifeartaigh |editor-link=Lochlainn O'Raifeartaigh |title=General Relativity: Papers in honour of J. L. Synge |publisher=Clarenden Press |location=Oxford |isbn=0-19-851126-4 |page=85 }}</ref> in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.<ref>{{cite book |last1=Fatibene |first1=L. |last2=Francaviglia |first2=M. |author-link2=Mauro Francaviglia |year=2003 |title=Natural and Gauge Natural Formalism for Classical Field Theories |publisher=Kluwer Academic |location=Dordrecht }}</ref>

In a given [[spin manifold]], that is in a Riemannian manifold <math>(M,g)</math> admitting a [[spin structure]], the Lie derivative of a [[spinor]] [[Field (mathematics)|field]] <math>\psi</math> can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the [[André Lichnerowicz]]'s local expression given in 1963:<ref>{{cite journal |last=Lichnerowicz |first=A. |year=1963 |title=Spineurs harmoniques |journal=C. R. Acad. Sci. Paris |volume=257 |pages=7–9 }}</ref>

:<math>\mathcal{L}_X \psi := X^{a}\nabla_{a}\psi - \frac14\nabla_{a}X_{b} \gamma^{a}\gamma^{b}\psi\, ,</math>

where <math>\nabla_{a}X_{b} = \nabla_{[a}X_{b]}</math>, as <math>X = X^{a}\partial_{a}</math> is assumed to be a [[Killing vector field]], and <math>\gamma^{a}</math> are [[Dirac matrices]].

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a ''generic'' vector field <math>X</math>, but explicitly taking the antisymmetric part of <math>\nabla_{a}X_{b}</math> only.<ref name="autogenerated317" /> More explicitly, Kosmann's local expression given in 1972 is:<ref name="autogenerated317"/>

:<math>\mathcal{L}_X \psi := X^{a}\nabla_{a}\psi - \frac18\nabla_{[a}X_{b]}[\gamma^{a},\gamma^{b}]\psi\, = \nabla_X \psi - \frac14 (d X^\flat)\cdot \psi\, ,</math>

where <math>[\gamma^{a},\gamma^{b}]= \gamma^a\gamma^b - \gamma^b\gamma^a</math> is the commutator, <math>d</math> is [[exterior derivative]], <math>X^\flat = g(X, -)</math> is the dual 1 form corresponding to <math>X</math> under the metric (i.e. with lowered indices) and <math> \cdot </math> is Clifford multiplication.

It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the [[Connection (differential geometry)|connection]]. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on the [[spinor bundle]]. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.

To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article,<ref>{{cite book |last1=Fatibene |first1=L. |last2=Ferraris |first2=M. |last3=Francaviglia |first3=M. |last4=Godina |first4=M. |year=1996 |chapter=A geometric definition of Lie derivative for Spinor Fields |title=Proceedings of the 6th International Conference on Differential Geometry and Applications, August 28th–September 1st 1995 (Brno, Czech Republic) |editor-last=Janyska |editor-first=J. |editor2-last=Kolář |editor2-first=I. |editor3-last=Slovák |editor3-first=J. |publisher=Masaryk University |location=Brno |pages=549–558 |isbn=80-210-1369-9 |arxiv=gr-qc/9608003v1 |bibcode=1996gr.qc.....8003F }}</ref><ref>{{cite journal |last1=Godina |first1=M. |last2=Matteucci |first2=P. |year=2003 |title=Reductive G-structures and Lie derivatives |journal=[[Journal of Geometry and Physics]] |volume=47 |issue=1 |pages=66–86 |doi=10.1016/S0393-0440(02)00174-2 |arxiv=math/0201235 |bibcode=2003JGP....47...66G |s2cid=16408289 }}</ref> where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the [[Kosmann lift]].

As for the tensor counterpart, also for spinors the vanishing of the Lie derivative along a Killing vector implements on the spinor the symmetries encoded by that Killing vector. However, differently from tensors, from spinors it is possible to build bi-linear quantities (such as the velocity vector <math>\overline{\psi}\gamma^{a}\psi</math> or the spin axial-vector <math>\overline{\psi}\gamma^{a}\gamma^5\psi</math>) which are tensors. A natural question that now arises is whether the vanishing of the Lie derivative along a Killing vector of a spinor is equivalent to the vanishing of the Lie derivative along the same Killing vector of all the spinor bi-linear quantities. While a spinor that is Lie-invariant implies that all its bi-linear quantities are also Lie invariant, the converse is in general not true.<ref>{{cite journal | author=Luca Fabbri, Stefano Vignolo, Roberto Cianci | date= 2024 | title = Polar form of Dirac fields: implementing symmetries via Lie derivative | journal = Lett. Math. Phys. | volume =  114| issue= 1 | pages = 21| doi = 10.1007/s11005-024-01770-7 | arxiv = 2310.10678 | bibcode= 2024LMaPh.114...21F }}</ref>

===Covariant Lie derivative===
If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. it has horizontal and vertical components), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.

Now, if we're given a vector field ''Y'' over ''M'' (but not the principal bundle) but we also have a [[Connection (mathematics)|connection]] over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matches ''Y'' and its vertical component agrees with the connection. This is the covariant Lie derivative.

See [[connection form]] for more details.

===Nijenhuis–Lie derivative===
Another generalization, due to [[Albert Nijenhuis]], allows one to define the Lie derivative of a differential form along any section of the bundle Ω<sup>''k''</sup>(''M'', T''M'') of differential forms with values in the tangent bundle. If ''K''&nbsp;∈&nbsp;Ω<sup>''k''</sup>(''M'', T''M'') and α is a differential ''p''-form, then it is possible to define the interior product ''i''<sub>''K''</sub>α of ''K'' and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:
:<math>\mathcal{L}_K\alpha=[d,i_K]\alpha = di_K\alpha-(-1)^{k-1}i_K \, d\alpha.</math>

==History==
In 1931, [[Władysław Ślebodziński]] introduced a new differential operator, later called by [[David van Dantzig]] that of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms.

The Lie derivatives of general geometric objects (i.e., sections of [[natural bundle|natural fiber bundle]]s) were studied by [[Albert Nijenhuis|A. Nijenhuis]], Y. Tashiro and [[Kentaro Yano (mathematician)|K. Yano]].

For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940, [[Léon Rosenfeld]]<ref>{{cite journal |last=Rosenfeld |first=L. |date=1940 |title=Sur le tenseur d'impulsion-énergie |journal=Mémoires Acad. Roy. D. Belg. |volume=18 |issue=6 |pages=1–30 }}</ref>—and before him (in 1921) [[Wolfgang Pauli]]<ref>{{cite book |last=Pauli |first=W. |title=Theory of Relativity |date=1 July 1981 |publisher=Dover |location=New York |orig-year=1921 |isbn=978-0-486-64152-2 }} ''See section 23''</ref>—introduced what he called a ‘local variation’ <math>\delta^{\ast}A</math> of a geometric object <math>A\,</math> induced by an infinitesimal transformation of coordinates generated by a vector field <math>X\,</math>. One can easily prove that his <math>\delta^{\ast}A</math> is <math> - \mathcal{L}_X(A)\,</math>.

==See also==
* [[Covariant derivative]]
* [[Connection (mathematics)]]
* [[Frölicher–Nijenhuis bracket]]
* [[Geodesic]]
* [[Killing vector field|Killing field]]
* [[Derivative of the exponential map]]

==Notes==
{{Reflist|30em}}

==References==
* {{cite book |first1=Ralph |last1=Abraham |author-link=Ralph Abraham (mathematician) |first2=Jerrold E. |last2=Marsden |author-link2=Jerrold E. Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=0-8053-0102-X }} ''See section 2.2''.
* {{cite book |first=David |last=Bleecker |title=Gauge Theory and Variational Principles |year=1981 |publisher=Addison-Wesley |isbn=0-201-10096-7 |url-access=registration |url=https://archive.org/details/gaugetheoryvaria00blee_0 }} ''See Chapter 0''.
* {{cite book |first=Jürgen |last=Jost |author-link=Jürgen Jost |title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer |location=Berlin |isbn=3-540-42627-2 }} ''See section 1.6''.
* {{cite book |last1=Kolář |first1=I. |last2=Michor |first2=P. |last3=Slovák |first3=J. |title=Natural operations in differential geometry|publisher=Springer-Verlag|isbn=9783662029503|url=http://www.emis.de/monographs/KSM/index.html|year=1993}} Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
* {{cite book|author-link=Serge Lang |last=Lang |first=S.|title=Differential and Riemannian manifolds|publisher=Springer-Verlag|year=1995|isbn=978-0-387-94338-1}} For generalizations to infinite dimensions.
* {{cite book|author-link=Serge Lang |last=Lang |first=S.|title=Fundamentals of Differential Geometry|publisher=Springer-Verlag|year=1999|isbn=978-0-387-98593-0}} For generalizations to infinite dimensions.
* {{cite book|author-link=Kentaro Yano (mathematician) |last=Yano |first=K. |title=The Theory of Lie Derivatives and its Applications
|url=https://archive.org/details/theoryofliederiv029601mbp|publisher=North-Holland|year=1957|isbn=978-0-7204-2104-0}} Classical approach using coordinates.

==External links==
* {{springer|title=Lie derivative|id=p/l058560}}

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[[Category:Generalizations of the derivative]]