In the mathematical field of
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two
vector fields ''X'' and ''Y'' on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'' a third vector field denoted .
Conceptually, the Lie bracket is the derivative of ''Y'' along the
flow generated by ''X'', and is sometimes denoted ''
'' ("Lie derivative of Y along X"). This generalizes to the
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of any
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
along the flow generated by ''X''.
The Lie bracket is an R-
bilinear operation and turns the set of all
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
vector fields on the manifold ''M'' into an (infinite-dimensional)
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
.
The Lie bracket plays an important role in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, for instance in the
Frobenius integrability theorem
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric term ...
, and is also fundamental in the geometric theory of
nonlinear control systems.
[, ]nonholonomic system
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...
s; , feedback linearization
Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form
where x(t) \in \mathbb^n is the state, u_1(t), ...
.
Definitions
There are three conceptually different but equivalent approaches to defining the Lie bracket:
Vector fields as derivations
Each smooth vector field
on a manifold ''M'' may be regarded as a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
acting on smooth functions
(where
and
of class
) when we define
to be another function whose value at a point
is the
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
of ''f'' at ''p'' in the direction ''X''(''p''). In this way, each smooth vector field ''X'' becomes a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
on ''C''
∞(''M''). Furthermore, any derivation on ''C''
∞(''M'') arises from a unique smooth vector field ''X''.
In general, the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
of any two derivations
and
is again a derivation, where
denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:
:
Flows and limits
Let
be the
flow associated with the vector field ''X'', and let D denote the
tangent map derivative operator. Then the Lie bracket of ''X'' and ''Y'' at the point can be defined as the
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
:
:
This also measures the failure of the flow in the successive directions
to return to the point ''x'':
:
In coordinates
Though the above definitions of Lie bracket are
intrinsic
In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
(independent of the choice of coordinates on the manifold ''M''), in practice one often wants to compute the bracket in terms of a specific coordinate system
. We write
for the associated local basis of the tangent bundle, so that general vector fields can be written
and
for smooth functions
. Then the Lie bracket can be computed as:
:
If ''M'' is (an open subset of) R
''n'', then the vector fields ''X'' and ''Y'' can be written as smooth maps of the form
and
, and the Lie bracket
is given by:
:
where
and
are
Jacobian matrices (
and
respectively using index notation) multiplying the column vectors ''X'' and ''Y''.
Properties
The Lie bracket of vector fields equips the real vector space
of all vector fields on ''M'' (i.e., smooth sections of the tangent bundle
) with the structure of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, which means
• , • is a map
with:
*R-
bilinearity
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
*Anti-symmetry,