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Lie derivative

From Academic Kids

In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by

<math> [A,B] := \mathcal{L}_A B = - \mathcal{L}_B A<math>

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M. Looking at it the other way round, the diffeomorphism group of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory.

Contents

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 in the next section.

One might start by defining the Lie derivative in terms of the differential of a function. Thus, give a function <math>f:M\rightarrow \mathbb{R}<math> and a vector field X defined on M, one defines the Lie derivative of f at point <math>p\in M<math> as

<math>\mathcal{L}_Xf(p)=df(p)\, [X(p)]<math>

where <math>df<math> is the differential of f. That is, <math>df:M\rightarrow T^*M<math> is the 1-form given by

<math>df = \frac{\partial f} {\partial x^a} dx^a<math>.

Here, the <math>dx^a<math> are the basis vectors for the cotangent bundle <math>T^*M<math>. Thus, the notation <math>df(p)\, [X(p)]<math> means that the inner product of the differential of f (at point p in M) is being taken with the vector field X (at point p).

Alternately, one might start by showing that a smooth vector field X on M defines a one-parameter family of curves on M. That is, one shows that there exists a curve <math>\gamma(t)<math> on M such that

<math>\frac{d\gamma}{dt}(t)=X(\gamma(t))<math>

with <math>p=\gamma(0)<math> for any point p in M. The existence of solutions to this first-order ordinary differential equation is given by the Picard-Lindelöf theorem (more generally, one says the existence of such curves is given by the Frobenius theorem). One then defines the Lie derivative as

<math>\mathcal{L}_Xf(p)=\frac{d}{dt} f(\gamma(t)) \vert_{t=0}<math>.

A third possible definition of the Lie derivative can be gotten by first defining the Lie bracket of a pair of vector fields. One starts by noting that the basis vectors for the tangent manifold can be written as <math>\frac{\partial}{\partial x^a}<math>, and so a vector field, expressed in terms of a selected set of basis vectors is written as

<math>X=X^a \frac{\partial}{\partial x^a}<math>

One defines the Lie bracket <math>[X,Y]<math> of a pair of vector fields as

<math>[X,Y]=

X^a \frac{\partial Y^b}{\partial x^a} \frac{\partial}{\partial x^b} - Y^a \frac{\partial X^b}{\partial x^a} \frac{\partial}{\partial x^b} <math>

One then defines the Lie derivative of the vector field Y to be equal to the Lie bracket of X and Y, that is,

<math>\mathcal{L}_X Y = [X,Y]<math>.

Depending on which of the above are chosen as the definition of the Lie derivative, the others can be proven to be equivalent. Thus, for example, one may prove that, for a differentiable function f,

<math>\mathcal{L}_X (f) = df(X) = X(f)<math>

and that

<math>[X,Y]f = X(Y(f)) - Y(X(f))<math>.

We complete this section by noting the definition of the Lie derivative on a 1-form <math>\omega = \omega_a dx^a<math> is given by

<math>\mathcal{L}_X \omega =

\left(\frac{\partial \omega_b} {\partial x^a} X^a + \frac{\partial X^a} {\partial x^b} \omega_a \right) dx^b <math>.

Properties

The Lie derivative has a number of properties. Let <math>\mathcal{F}(M)<math> be the algebra of functions defined on the manifold M. Then

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

is a 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:

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

which is 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. 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 the Lie bracket, forms a Lie algebra.

Relationship to the exterior derivative

The Lie derivative is closely related to the exterior derivative and thus to Elie Cartan's theory of differential forms. Both attempt to capture the idea of a derivative, and the differences are almost notational in nature. These differences of notation can be bridged by introducing the idea of an antiderivation or equivalently an inner 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 <math>\omega \in \Lambda^{k+1}(M)<math> be a k+1-form. The inner product of X and ω is

<math>i_X\omega (X_1,\ldots,X_k) = \omega (X,X_1,\ldots,X_k)<math>

Note that

<math>i_X:\Lambda^{k+1}(M) \rightarrow \Lambda^k(M)<math>

and that <math>i_X<math> is a <math>\wedge<math>-antiderivation. 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 anther 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 = fi_X\omega<math>

The relationship between exterior derivatives and Lie derivatives can then be summarized in the following relationships. For an ordinary function f, the Lie derivative is just the contraction of the exterior derivative with the vector field 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>.

When ω is a 1-form, the above identity is frequently written as

<math>d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]).<math>

The derivative of products is distributed:

<math>\mathcal{L}_{fX}\omega =

f\mathcal{L}_X\omega + df \wedge i_X \omega<math>

Lie derivative of tensor fields

In differential geometry, if we have a differentiable tensor field <math>T<math> of rank <math>(p,q)<math> (which we consider as a differentiable linear map of smooth sections <math>\alpha, \beta, \ldots<math> of the cotangent bundle <math>T^*M<math> and of sections <math>X, Y, \ldots<math> of the tangent bundle <math>TM<math>, written <math>T (\alpha, \beta, \ldots, X, Y, \ldots )<math>, such that for any collection of smooth functions <math>f_1,\ldots,f_p,f_{p+1},\ldots,f_{p+q}<math> we have

<math>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)<math>),

and if further we have a differentiable vector field (i.e. a smooth section of the tangent bundle) <math>A<math>, then the linear map

<math>(\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<math>

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 <math>T<math> with respect to <math>A<math>.

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

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

<math>\mathcal{L}_U T = \frac{d}{dt}\left(\psi^*_t T\right) \vert_{\psi(t)=p}<math>.

See also

References

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