Lie bracket of vector fields

See Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 for more on the definition of the Lie bracket and Lie derivative

Lie derivative

In mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one vector field along the flow of another vector field.The Lie derivative is a derivation on the algebra of tensor fields over a manifold M...

 for the derivation

In the mathematical field of differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, the Lie bracket of vector fields,Jacobi–Lie bracket, or Commutator of vector fields is a bilinear

Bilinear

Bilinear may refer to* Bilinear sampling, a method in computer graphics for choosing the color of a texture.* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...

 differential operator

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, accepting a function and returning another .There are certainly reasons not to restrict...

 which assigns, to any two vector field

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...

s X and Y on a smooth manifold M, a third vector field denoted [X, Y]. It is closely related to, and sometimes also known as, the Lie derivative

Lie derivative

In mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one vector field along the flow of another vector field.The Lie derivative is a derivation on the algebra of tensor fields over a manifold M...

. In particular, the bracket equals the Lie derivative .

It plays an important role in differential geometry and differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, and is also fundamental in the geometric theory for nonlinear control systems ' onMouseout='HidePop("22391")' href="http://www.absoluteastronomy.com/topics/Nonholonomic_system">nonholonomic system

Nonholonomic system

A nonholonomic system in physics and mathematics, is a system whose state depends on the path taken to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space, but finally returns to...

s; , feedback linearization

Feedback linearization

Feedback linearization is a common approach used in controlling nonlinear systems. The approach involves coming up with a transformation of the nonlinear system into an equivalent linear system through a change of variables and a suitable control input...

).

A generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.

Definition

Let X and Y be smooth vector fields on a smooth -manifold M. The Jacobi–Lie bracket or simply Lie bracket of X and Y, denoted is the unique vector field such that
where is the Lie derivative

Lie derivative

In mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one vector field along the flow of another vector field.The Lie derivative is a derivation on the algebra of tensor fields over a manifold M...

 with respect to the vector field X. For a finite-dimensional manifold M we can define the Jacobi–Lie bracket in local coordinates as
where n is the dimension of M.

The Lie bracket of vector fields equips the real vector space (i.e., smooth sections of the tangent bundle of ) with the structure of a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

, i.e., [.,.] is a map from VV to V with the following properties

• [.,.] is R-bilinear
• This is the Jacobi identity
  Jacobi identity
  In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation...
  
  .
• For functions f and g we have

An immediate consequence of the second property is that for any .

The name commutator is used because of the following fact:

Theorem:

iff their corresponding flows commute (i.e. ).

Examples

For a matrix Lie group, smooth vector fields can be locally represented in the corresponding Lie algebra. Since the Lie algebra associated with a Lie group is isomorphic to the group's tangent space at the identity, elements of the Lie algebra of a matrix Lie group are also matrices. Hence the Jacobi–Lie bracket corresponds to the usual commutator for a matrix group:
where juxtaposition indicates matrix multiplication.

Applications

The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.