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 ''M'' a third vector field denoted .
Conceptually, the Lie bracket is the derivative of ''Y'' along the
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
generated by ''X'', and is sometimes denoted ''
'' ("Lie derivative of Y along X"). This generalizes to the
Lie derivative 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 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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.
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 mult ...
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, 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 constraint ...
s; , feedback linearization.
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 retur ...
acting on smooth functions
(where
and
of class
) when we define
to be another function whose value at a point
is the
directional derivative of ''f'' at ''p'' in the direction ''X''(''p''). In this way, each smooth vector field ''X'' becomes a
derivation 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, ...
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
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
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:
:
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 (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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
, which means
• , • is a map
with:
*R-
bilinearity
*Anti-symmetry,