Lie derivative

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In mathematics, a Lie derivative, named after Sophus Lie by Władysław Ślebodziński, is a derivation on the algebra of tensor fields 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

[A,B] = \mathcal{L}_A B=-\mathcal{L}_B A

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

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[edit] 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.

[edit] 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\rightarrow \mathbb{R} and a vector field X defined on M, one defines the Lie derivative of f at point p\in M as

\mathcal{L}_Xf(p)=X_p(f)=\nabla_Xf(p)

the usual derivative of f along the vector field X.

In fancier terms, this can be restated using the dual pairing between the tangent bundle and cotangent bundle of M as follows:

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

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

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

Here, the dxa are the basis vectors for the cotangent bundle T * M. (The Einstein summation convention is implied in the formula.) Thus, the notation df(p)\, [X(p)] 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). Writing X in the xa coordinates,

X=X^a\frac{\partial}{\partial x^a}

we have

\mathcal{L}_Xf(p)=df(p)\, [X(p)]=X^a\frac{\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 γ(t) on M such that

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

with p = γ(0). 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

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

[edit] 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 the Lie 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 xa be coordinates on M. One starts by noting that the basis vectors for the tangent 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}\right) \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 \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}}

[edit] The Lie derivative of differential forms

The Lie derivative can also be defined on differential forms. In this context, it 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 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_X\omega) (X_1, \ldots, X_k) = \omega (X,X_1, \ldots, X_k)\,

Note that

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

and that iX is a \wedge-antiderivation. That is, iX 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_X\omega

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

[edit] Properties

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

\mathcal{L}_X : \mathcal{F}(M) \rightarrow \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 + f\mathcal{L}_Xg.

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

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

which is may also be written in the equivalent notation

\mathcal{L}_X(f\otimes Y)= (\mathcal{L}_Xf) \otimes Y + f\otimes \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 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

  • \mathcal{L}_X(\alpha\wedge\beta)=(\mathcal{L}_X\alpha)\wedge\beta+\alpha\wedge(\mathcal{L}_X\beta)
  • [\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
  • [\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.

[edit] Lie derivative of tensor fields

More generally, if we have a differentiable tensor field T of rank (p,q) and a differentiable vector field Y (i.e. a differentiable section of the tangent 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 the vector flow of Y and denote φt(p) := φ(p, t). For each sufficiently small t, φt is a diffeomorphism from an neighborhood in M to another neighborhood in M, and φ0 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}\right|_{t=0}\left((\phi_t)_*T_{\phi_{-t}(p)}\right).

where (φt)* 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)=\nabla_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}_Y\alpha=i_Yd\alpha+di_Y\alpha.
Axiom 4. The Lie derivative obeys the Leibniz rule. For any tensor fields S and T, we have
\mathcal{L}_Y(S\otimes T)=(\mathcal{L}_YS)\otimes T+S\otimes (\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 α1, α2, …, αq of the cotangent bundle T*M and of sections X1, X2, … Xp of the tangent bundle TM, written T(α1, α2, …, X1, X2, …) 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}_Y\alpha_1, \alpha_2, \ldots, X_1, X_1, \ldots) - T(\alpha_1, \mathcal{L}_Y\alpha_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.

[edit] Coordinate expressions

Let xa be a system of coordinates. 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(\nabla_cT^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) - (\nabla_cX ^{a_1}) T ^{c \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\nabla_cX^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} +
+ (\nabla_{b_1}X^c) T ^{a_1 \ldots a_r}{}_{c \ldots b_s} + \ldots + (\nabla_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c}

here, the notation ∇ means taking the gradient in the x 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.

[edit] Generalizations

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

[edit] Nijenhuis-Lie derivative

This article has defined the usual Lie derivative of a differential form along a vector field. One generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any contravariant tensor field. In detail, if K is a contravariant tensor and α is a differential p-form, then it is possible define the interior product iKα of K and α. The Nijenhuis-Lie derivative is then the anticommutator of the interior product and the exterior derivative:

\mathcal{L}_K\alpha=di_K\alpha+i_Kd\alpha.

The Nijenhuis-Lie derivative enjoys many algebraic properties similar to those of the Lie derivative, with one notable exception: it is not a derivation in the usual sense.

[edit] See also

[edit] References

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