- Lie derivative
mathematics, the Lie derivative, named after Sophus Lieby Władysław Ślebodziński, evaluates the change of one vector fieldalong the flow of another vector field.
The Lie derivative is a derivation on the algebra of
tensor fields over a manifold"M". The vector spaceof all Lie derivatives on "M" forms an infinite dimensional Lie algebrawith respect to the Lie bracketdefined by
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 diffeomorphism group of "M" has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie grouptheory.
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 and a
vector field"X" defined on "M", one defines the Lie derivative of "f" at point as
directional derivativeof "f" along the vector field "X".
In fancier terms, this can be restated using the dual pairing between the
tangent bundleand cotangent bundleof "M" as follows:
where is the differential of "f". That is, is the
Here, the are the
basis vectors for the tangent bundleand the is the dual basisin the cotangent bundle. (The Einstein summation conventionis implied in the formula.) Thus, the notation means that the inner productof 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,
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
curveon "M" such that
with . The existence of solutions to this first-order
ordinary differential equationis 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:.
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 bracketof 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, :.
The first definition of the Lie bracket uses the local coordinate expressions of the vector fields "X" and "Y". Let "x""a" be
coordinateson "M". One starts by noting that the basis vectors for the tangent bundlecan be written as , and so a vector field, expressed in terms of this selected set of basis vectors is written as :One defines the Lie bracket of a pair of vector fields as:
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::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:::
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 be a "k"+1-form. The interior product of "X" and ω is
: and that is a -antiderivation. That is, is R-linear, and
for and η another differential form. Also, for a function , that is a real or complex-valued function on "M", one has
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":
For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in "X":
The derivative of products is distributed:
The Lie derivative has a number of properties. Let be the
algebraof functions defined on the manifold"M". Then
is a derivation on the algebra . That is, is R-linear and
Similarly, it is a derivation on where is the set of vector fields on "M":
which is may also be written in the equivalent notation
tensor productsymbol 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,
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
* where "i" denotes interior multiplication between vector fields and differential forms.
Lie derivative of tensor fields
More generally, if we have a
differentiabletensor field "T" of rank 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 flowof "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 φ0 is the identity diffeomorphism. The Lie derivative of "T" is defined at a point "p" by
where (φ"t")* is the pushforward along the diffeomorphism. In other words, if you have a tensor field and an infinitesimal generator of a diffeomorphism given by a vector field "Y", then 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::
:Axiom 2. The Lie derivative of a vector field is the Lie bracket. So if "X" is a vector field, one has::
: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,::
:Axiom 4. The Lie derivative obeys the Leibniz rule. For any tensor fields "S" and "T", we have::
Explicitly, let "T" be a tensor field of type ("p","q"). Consider "T" to be a differentiable
multilinear mapof smooth sections α1, α2, ..., αq of the cotangent bundle "T*M" and of sections "X"1, "X"2, ... "X"p of the tangent bundle"TM", written "T"(α1, α2, ..., "X"1, "X"2, ...) into R. Define the Lie derivative of "T" along "Y" by the formula
The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the
Leibniz rulefor differentiation.
coordinatenotation, for a type (r,s) tensor field , the Lie derivative along is :::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.:::Notice the new term at the end of the expression.
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
connectionover 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.
connection formfor more details.
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::
* Killing field
Ralph Abrahamand Jerrold 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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