- Lie bracket of vector fields
:"See
Lie algebra for more on the definition of the Lie bracket andLie derivative for the derivationIn the mathematical field of
differential topology , the Lie bracket of vector fields or Jacobi–Lie bracket is abilinear differential operator which assigns, to any twovector field s "X" and "Y" on asmooth manifold "M", a third vector field denoted ["X", "Y"] . It is closely related to, and sometimes also known as, theLie derivative . In particular, the bracket X,Y] equals the Lie derivative mathcal{L}_X Y.It plays an important role in
differential geometry anddifferential topology , and is also fundamental in the geometric theory for nonlinear control systems.Definition
Let "X" and "Y" be smooth vector fields on a smooth n-manifold "M". The Jacobi-Lie bracket or simply Lie bracket of "X" and "Y", denoted X,Y] is the unique vector field such that
:mathcal{L}_{ [X,Y] } = mathcal{L}_X circ mathcal{L}_Y - mathcal{L}_Y circ mathcal{L}_X
where mathcal{L}_X is the
Lie derivative with respect to the vector field "X." For a finite-dimensional manifold "M" we can define the Jacobi-Lie bracket in local coordinates as:X,Y] ^i= sum_{j=1}^n left (X^j frac {partial Y^i}{partial x^j} ight ) - left ( Y^j frac {partial X^i}{partial x^j} ight )
where "n" is the dimension of "M."
The Lie bracket of vector fields equips the real vector space V=Gamma^{infty}(TM) (i.e., smooth sections of the tangent bundle of M) with the structure of a
Lie algebra , i.e., [.,.] is a map from VimesV to V with the following properties
* [.,.] is R-bilinear
*X,Y] =- [Y,X] ,
*X, [Y,Z] + [Z, [X,Y] + [Y, [Z,X] =0., This is theJacobi identity .An immediate consequence of these properties is that X,X] =0 for any X.
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:
:X,Y] = XY - YX
where juxtaposition indicates matrix multiplication.
Applications
The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.
References
* Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
* For generalizations to infinite dimensions.
*
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