Lie bracket of vector fields

:"See Lie algebra for more on the definition of the Lie bracket and Lie derivative for the derivation

In the mathematical field of differential topology, the Lie bracket of vector fields or Jacobi–Lie bracket is a bilinear differential operator which assigns, to any two vector fields "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. In particular, the bracket $[X,Y]$ equals the Lie derivative $mathcal\{L\}\_X\; Y$.

It plays an important role in differential geometry and differential 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 V$imes$V 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 the Jacobi 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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