# Lie derivative

In

mathematics , the**Lie derivative**, named afterSophus Lie byWładysław Ślebodziński , evaluates the change of onevector field along the flow of another vector field.The Lie derivative is a derivation on the algebra of

tensor field s over amanifold "M". Thevector space of all Lie derivatives on "M" forms an infinite dimensionalLie algebra with respect to theLie bracket defined by:$[A,B]\; =\; mathcal\{L\}\_A\; B\; =\; -mathcal\{L\}\_B\; A.$

The Lie derivatives are represented by vector fields, as

infinitesimal generator s of flows (activediffeomorphism s) on "M". Looking at it the other way around, the diffeomorphism group of "M" has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to theLie group theory.**Definition**The Lie derivative may be defined in several equivalent ways. In this section, to keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields. The Lie derivative can also be defined to act on general tensors, as developed later in the article.

**The Lie derivative of a function**One might start by defining the Lie derivative in terms of the differential of a function. Thus, given a function $f:M\; ightarrow\; mathbb\{R\}$ and a

vector field "X" defined on "M", one defines the Lie derivative of "f" at point $pin\; M$ as:$mathcal\{L\}\_Xf(p)=X\_p(f)=\; abla\_Xf(p)$

the

directional derivative of "f" along the vector field "X".In fancier terms, this can be restated using the dual pairing between the

tangent bundle andcotangent bundle of "M" as follows::$mathcal\{L\}\_Xf(p)=df(p),\; [X(p)]$

where $df$ is the differential of "f". That is, $df:M\; ightarrow\; T^*M$ is the

1-form given by:$df\; =\; frac\{partial\; f\}\; \{partial\; x\_a\}\; dx^a.$

Here, the $dx\_a$ are the

basis vector s for thetangent bundle $TM$ and the $dx^a$ is thedual basis in thecotangent bundle $T^*M$. (TheEinstein summation convention is implied in the formula.) Thus, the notation $df(p),\; [X(p)]$ means that theinner product of the differential of "f" (at point "p" in "M") is being taken with the vector field "X" (at point "p"). Writing "X" in the "x"^{"a"}coordinates ,:$X=X^afrac\{partial\}\{partial\; x\_a\}$

we have

:$mathcal\{L\}\_Xf(p)=df(p),\; [X(p)]\; =X^afrac\{partial\; f\}\{partial\; x\_a\}$

which recovers the original definition of the Lie derivative of a function.

Alternately, one might start by showing that a smooth vector field "X" on "M" defines a family of curves on "M". That is, one shows that for any point "p" in "M" there exists a

curve $gamma(t)$ on "M" such that:$frac\{dgamma\}\{dt\}(t)=X(gamma(t))$

with $p=gamma(0)$. The existence of solutions to this first-order

ordinary differential equation is given by thePicard-Lindelöf theorem (more generally, one says the existence of such curves is given by the Frobenius theorem). One then defines the Lie derivative as:$mathcal\{L\}\_Xf(p)=frac\{d\}\{dt\}\; f(gamma(t))\; vert\_\{t=0\}$.**The Lie derivative of a vector field**The Lie derivative of a function has now been defined in several ways. In each case, the Lie derivative of a function agrees with the usual idea of differentiation along a vector field from

multivariable calculus . The Lie derivative can be defined for vector fields by first defining theLie bracket of a pair of vector fields "X" and "Y", denoted ["X","Y"] . There are several approaches to defining the Lie bracket, all of which are equivalent. Regardless of the chosen definition, one then defines the Lie derivative of the vector field "Y" to be equal to the Lie bracket of "X" and "Y", that is, :$mathcal\{L\}\_X\; Y\; =\; [X,Y]$.The first definition of the Lie bracket uses the local coordinate expressions of the vector fields "X" and "Y". Let "x"

_{"a"}becoordinates on "M". One starts by noting that the basis vectors for thetangent bundle can be written as $frac\{partial\}\{partial\; x\_a\}$, and so a vector field, expressed in terms of this selected set of basis vectors is written as :$X=X^a\; frac\{partial\}\{partial\; x\_a\}$One defines the**Lie bracket**$[X,Y]$ of a pair of vector fields as:$[X,Y]\; :=(X(Y^a)\; -\; Y(X^a))\; frac\{partial\}\{partial\; x\_a\}\; =left(X^b\; frac\{partial\; Y^a\}\{partial\; x\_b\}\; -\; Y^b\; frac\{partial\; X^a\}\{partial\; x\_b\}\; ight)\; frac\{partial\}\{partial\; x\_a\}$The second definition is intrinsic in that it does not rely on the use of coordinates. Since a vector field can be identified with a first-order differential operator on functions, the Lie bracket of two vector fields can be defined as follows. If "X" and "Y" are two vector fields, then the Lie bracket of "X" and "Y" is also a vector field, denoted by ["X","Y"] , defined by the equation::$[X,Y]\; (f)\; :=\; X(Y(f))-Y(X(f))\; ,.$Using a local coordinate expression for "X" and "Y", one can prove that this is equivalent to the previous definition of the Lie bracket.

Other equivalent definitions are::$(mathcal\{L\}\_X\; Y)\_x\; :=\; lim\_\{t\; o\; 0\}\; (mathrm\{T\}(mathrm\{Fl\}^X\_\{-t\})\; Y\_\{mathrm\{Fl\}^X\_t(x)\}\; -\; Y\_x)/t\; =\; left.frac\{mathrm\{d\{mathrm\{d\}\; t\}\; ight|\_\{t=0\}\; mathrm\{T\}(mathrm\{Fl\}^X\_\{-t\})\; Y\_\{mathrm\{Fl\}^X\_t(x)\}$:$mathcal\{L\}\_X\; Y\; :=\; left.frac\{mathrm\{d\}^2\}\{2mathrm\{d\}^2\; t\}\; ight|\_\{t=0\}\; mathrm\{Fl\}^Y\_\{-t\}\; circ\; mathrm\{Fl\}^X\_\{-t\}\; circ\; mathrm\{Fl\}^Y\_\{t\}\; circ\; mathrm\{Fl\}^X\_\{t\}\; =\; left.frac\{mathrm\{d\{mathrm\{d\}\; t\}\; ight|\_\{t=0\}\; mathrm\{Fl\}^Y\_\{-sqrt\{t\; circ\; mathrm\{Fl\}^X\_\{-sqrt\{t\; circ\; mathrm\{Fl\}^Y\_\{sqrt\{t\; circ\; mathrm\{Fl\}^X\_\{sqrt\{t$

**The Lie derivative of differential forms**The Lie derivative can also be defined on

differential forms . In this context, it is closely related to theexterior 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**antiderivation**or equivalently an interior product, after which the relationships fall out as a set of identities.Let "M" be a manifold and "X" a vector field on "M". Let $omega\; in\; Lambda^\{k+1\}(M)$ be a "k"+1-form. The

**interior product**of "X" and ω is:$(i\_Xomega)\; (X\_1,\; ldots,\; X\_k)\; =\; omega\; (X,X\_1,\; ldots,\; X\_k),$

Note that

:$i\_X:Lambda^\{k+1\}(M)\; ightarrow\; Lambda^k(M)\; ,$ and that $i\_X$ is a $wedge$-antiderivation. That is, $i\_X$ is

**R**-linear, and:$i\_X\; (omega\; wedge\; eta)\; =\; (i\_X\; omega)\; wedge\; eta\; +\; (-1)^k\; omega\; wedge\; (i\_X\; eta)$

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

:$i\_\{fX\}\; omega\; =\; f,i\_Xomega$

The relationship between

exterior derivative s 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":

:$mathcal\{L\}\_Xomega\; =\; i\_Xdomega\; +\; d(i\_X\; omega)$.

The derivative of products is distributed:

:$mathcal\{L\}\_\{fX\}omega\; =\; fmathcal\{L\}\_Xomega\; +\; df\; wedge\; i\_X\; omega$

**Properties**The Lie derivative has a number of properties. Let $mathcal\{F\}(M)$ be the

algebra of functions defined on themanifold "M". Then:$mathcal\{L\}\_X\; :\; mathcal\{F\}(M)\; ightarrow\; mathcal\{F\}(M)$

is a derivation on the algebra $mathcal\{F\}(M)$. That is,$mathcal\{L\}\_X$ is

**R**-linear and:$mathcal\{L\}\_X(fg)=(mathcal\{L\}\_Xf)\; g\; +\; fmathcal\{L\}\_Xg$.

Similarly, it is a derivation on $mathcal\{F\}(M)\; imes\; mathcal\{X\}(M)$ where $mathcal\{X\}(M)$ is the set of vector fields on "M":

:$mathcal\{L\}\_X(fY)=(mathcal\{L\}\_Xf)\; Y\; +\; fmathcal\{L\}\_X\; Y$

which is may also be written in the equivalent notation

:$mathcal\{L\}\_X(fotimes\; Y)=(mathcal\{L\}\_Xf)\; otimes\; Y\; +\; fotimes\; mathcal\{L\}\_X\; Y$

where the

tensor product symbol $otimes$ 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 . Thus, for example, considered as a derivation on a vector field,:$mathcal\{L\}\_X\; [Y,Z]\; =\; [mathcal\{L\}\_X\; Y,Z]\; +\; [Y,mathcal\{L\}\_X\; Z]$

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 aLie 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

* $mathcal\{L\}\_X(alphawedgeeta)=(mathcal\{L\}\_Xalpha)wedgeeta+alphawedge(mathcal\{L\}\_Xeta)$

* $[mathcal\{L\}\_X,mathcal\{L\}\_Y]\; alpha:=mathcal\{L\}\_Xmathcal\{L\}\_Yalpha-mathcal\{L\}\_Ymathcal\{L\}\_Xalpha=mathcal\{L\}\_\{\; [X,Y]\; \}alpha$

* $[mathcal\{L\}\_X,i\_Y]\; alpha=\; [i\_X,mathcal\{L\}\_Y]\; alpha=i\_\{\; [X,Y]\; \}alpha,$ where "i" denotes interior multiplication between vector fields and differential forms.**Lie derivative of tensor fields**More generally, if we have a

differentiable tensor field "T" of rank $(p,q)$ and a differentiablevector field "Y" (i.e. a differentiable section of thetangent bundle "TM"), then we can define the Lie derivative of "T" along "Y". Let φ:"M"×**R**→"M" be the one-parameter semigroup of local diffeomorphisms of "M" induced by thevector flow of "Y" and denote φ_{"t"}("p") := φ("p", "t"). For each sufficiently small "t", φ_{"t"}is a diffeomorphism from an neighborhood in "M" to another neighborhood in "M", and φ_{0}is the identity diffeomorphism. The Lie derivative of "T" is defined at a point "p" by:$(mathcal\{L\}\_Y\; T)\_p=left.frac\{d\}\{dt\}\; ight|\_\{t=0\}left((phi\_t)\_*T\_\{phi\_\{-t\}(p)\}\; ight)$.

where (φ

_{"t"})_{*}is the pushforward along the diffeomorphism. In other words, if you have a tensor field $T$ and an infinitesimal generator of a diffeomorphism given by a vector field "Y", then $mathcal\{L\}\_\{Y\}\; T$ is nothing other than the infinitesimal change in "T" under the infinitesimal diffeomorphism.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 the directional derivative of the function. So if "f" is a real valued function on "M", then::$mathcal\{L\}\_Yf=Y(f)=\; abla\_Y\; f.$:

**Axiom 2.**The Lie derivative of a vector field is the Lie bracket. So if "X" is a vector field, one has::$mathcal\{L\}\_YX=\; [Y,X]\; .$:

**Axiom 3.**The Lie derivative of a differential form is the anticommutator of the interior product with the exterior derivative. So if α is a differential form,::$mathcal\{L\}\_Yalpha=i\_Ydalpha+di\_Yalpha.$:

**Axiom 4.**The Lie derivative obeys the Leibniz rule. For any tensor fields "S" and "T", we have::$mathcal\{L\}\_Y(Sotimes\; T)=(mathcal\{L\}\_YS)otimes\; T+Sotimes\; (mathcal\{L\}\_YT).$Explicitly, let "T" be a tensor field of type ("p","q"). Consider "T" to be a differentiable

multilinear map of smooth sections α^{1}, α^{2}, ..., α^{q}of the cotangent bundle "T*M" and of sections "X"_{1}, "X"_{2}, ... "X"_{p}of thetangent bundle "TM", written "T"(α^{1}, α^{2}, ..., "X"_{1}, "X"_{2}, ...) into**R**. Define the Lie derivative of "T" along "Y" by the formula:$(mathcal\{L\}\_Y\; T)(alpha\_1,\; alpha\_2,\; ldots,\; X\_1,\; X\_2,\; ldots)\; =Y(T(alpha\_1,alpha\_2,ldots,X\_1,X\_2,ldots))$::$-\; T(mathcal\{L\}\_Yalpha\_1,\; alpha\_2,\; ldots,\; X\_1,\; X\_1,\; ldots)\; -\; T(alpha\_1,\; mathcal\{L\}\_Yalpha\_2,\; ldots,\; X\_1,\; X\_1,\; ldots)\; -ldots$::$-\; T(alpha\_1,\; alpha\_2,\; ldots,\; mathcal\{L\}\_YX\_1,\; X\_2,\; ldots)\; -\; T(alpha\_1,\; alpha\_2,\; ldots,\; X\_1,\; mathcal\{L\}\_YX\_2,\; ldots)\; -\; ldots$

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the

Leibniz rule for differentiation.**Coordinate expressions**In

coordinate notation, for a type (r,s) tensor field $T$, the Lie derivative along $X$ is :$mathcal\; L\_X\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_s\}\; =\; X^c(\; abla\_c\; T^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_s\})\; -\; (\; abla\_c\; X\; ^\{a\_1\})\; T\; ^\{c\; a\_2\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_s\}\; -\; ldots\; -\; (\; abla\_c\; X^\{a\_r\})\; T\; ^\{a\_1\; ldots\; a\_\{r-1\}c\}\{\}\_\{b\_1\; ldots\; b\_s\}\; +$::$+\; (\; abla\_\{b\_1\}\; X^c)\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{c\; b\_2\; ldots\; b\_s\}\; +\; ldots\; +\; (\; abla\_\{b\_s\}X^c)\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_\{s-1\}\; c\}$here, the notation ∇ means taking the gradient (i.e. partial derivative). The Lie derivative of a tensor is another tensor of the same type. The above formula gives the same resulting tensor in any coordinate system.Alternatively, if we are using a torsion-free connection, then ∇ could also mean the

covariant derivative . For a torsion-free connection, both definitions are equivalent.The definition can be extended further to tensor densities of weight "w" for any real "w". If "T" is such a tensor density, then its Lie derivative is a tensor density of the same type and weight.:$mathcal\; \{L\}\_X\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_s\}\; =\; X^c(\; abla\_c\; T^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_s\})\; -\; (\; abla\_c\; X\; ^\{a\_1\})\; T\; ^\{c\; a\_2\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_s\}\; -\; ldots\; -\; (\; abla\_c\; X^\{a\_r\})\; T\; ^\{a\_1\; ldots\; a\_\{r-1\}c\}\{\}\_\{b\_1\; ldots\; b\_s\}\; +$::$+\; (\; abla\_\{b\_1\}\; X^c)\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{c\; b\_2\; ldots\; b\_s\}\; +\; ldots\; +\; (\; abla\_\{b\_s\}\; X^c)\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_\{s-1\}\; c\}\; +\; w\; (\; abla\_\{c\}\; X^c)\; T\; ^\{a\_1\; ldots\; a\_r\}\{\}\_\{b\_1\; ldots\; b\_\{s$Notice the new term at the end of the expression.

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

**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 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^{"k"}("M", T"M") of differential forms with values in the tangent bundle. If "K" ∈ Ω^{"k"}("M", T"M") and α is a differential "p"-form, then it is possible define the interior product "i"_{"K"}α of "K" and α. The Nijenhuis-Lie derivative is then the anticommutator of the interior product and the exterior derivative::$mathcal\{L\}\_Kalpha=\; [d,i\_K]\; alpha\; =\; di\_Kalpha-(-1)^\{k-1\}i\_Kdalpha.$**ee also*** Killing field

*Geodesic

*Covariant derivative

*Connection (mathematics)

*Frölicher-Nijenhuis bracket **References***

Ralph Abraham andJerrold E. Marsden , "Foundations of Mechanics", (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X "See section 2.2".

* David Bleecker, "Gauge Theory and Variational Principles", (1981), Addison-Wesley Publishing, ISBN 0-201-10096-7. "See Chapter 0".

* Jurgen Jost, "Riemannian Geometry and Geometric Analysis", (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 "See section 1.6".

* Extensive discussion of Lie brackets, and the general theory of Lie derivatives.

* For generalizations to infinite dimensions.

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