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| {{Use dmy dates|date=October 2011}}
| | I'm Yoshiko Oquendo. Kansas is our birth place and my mothers and fathers reside nearby. Interviewing is what she does in her day occupation but soon her spouse and her will start their own company. Climbing is what adore doing.<br><br>Here is my page [http://529Design.net/UserProfile/tabid/61/userId/27197/Default.aspx extended car warranty] |
| In the mathematical field of [[differential topology]], the '''Lie bracket of vector fields''', also known as the '''Jacobi–Lie bracket''' or the '''commutator of vector fields''', is an operator which assigns, to any two [[vector field]]s ''X'' and ''Y'' on a [[smooth manifold]] ''M'', a third vector field denoted [''X'', ''Y''].
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| Conceptually, the Lie bracket [''X'',''Y''] is the derivative of ''Y'' in the `direction' generated by ''X''. It is a special case of the [[Lie derivative]] which allows to form the derivative of any [[tensor field]] in the direction generated by ''X''. Indeed, [''X,Y''] equals the Lie derivative <math>\mathcal{L}_X Y</math>.
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| The Lie bracket is an '''R'''-[[bilinear operator|bilinear]] operation and turns the set of all vector fields on the manifold ''M'' into an (infinite-dimensional) [[Lie algebra]].
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| The Lie bracket plays an important role in [[differential geometry]] and [[differential topology]], for instance in the [[Frobenius theorem (differential topology)|Frobenius theorem]], and is also fundamental in the geometric theory for [[nonlinear control theory|nonlinear control systems]] ({{harvnb|Isaiah|2009|pp=20–21}}, [[nonholonomic system]]s; {{harvnb|Khalil|2002|pp=523–530}}, [[feedback linearization]]).
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| ==Definitions==
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| There are three conceptually different but equivalent approaches to defining the Lie bracket:
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| ===Vector fields as derivations===
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| Each vector field ''X'' on a smooth manifold ''M''
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| may be regarded as a [[differential operator]] acting on smooth
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| functions on ''M''. Indeed, each
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| smooth vector field ''X'' becomes a [[Derivation (abstract algebra)|derivation]] on the smooth
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| functions ''C<sup>∞</sup>(M)'' when we define ''X(f)'' to be the element of ''C<sup>∞</sup>(M)'' whose value at a point ''p'' is the [[directional derivative]] of ''f'' at ''p'' in the direction ''X(p)''. Furthermore, it is known that any derivation on ''C<sup>∞</sup>(M)'' arises in this fashion from a uniquely determined smooth vector field ''X''.
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| In general, the [[commutator]] <math>\delta_1\circ \delta_2 - \delta_2\circ\delta_1</math> of any two derivations <math>\delta_1</math> and <math>\delta_2</math> is again a derivation. This can be used to define the Lie bracket of vector fields as follows.
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| The Lie bracket, [''X,Y''], of two smooth vector fields
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| ''X'' and ''Y'' is the smooth vector field [''X,Y''] such that
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| :<math>[X,Y](f) = X(Y(f))-Y(X(f)) \;\;\text{ for all } f\in C^\infty(M).</math>
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| ===Flows and limits===
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| Let <math>\Phi^X_t</math> be the [[Flow (mathematics)|flow]] associated with the vector field ''X'', and let d denote the [[Pushforward (differential)|tangent map derivative operator]]. Then the Lie bracket of ''X'' and ''Y'' at the point ''x''∈''M'' can be defined as
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| :<math>[X, Y]_x := \lim_{t \to 0}\frac{(\mathrm{d}\Phi^X_{-t}) Y_{\Phi^X_t(x)} - Y_x}t = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\mathrm{d}\Phi^X_{-t}) Y_{\Phi^X_t(x)}</math>
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| or equivalently
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| :<math>[X, Y]_x := \left.\frac12\frac{\mathrm{d}^2}{\mathrm{dt}^2}\right|_{t=0} (\Phi^Y_{-t} \circ \Phi^X_{-t} \circ \Phi^Y_{t} \circ \Phi^X_{t})(x) = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\Phi^Y_{-\sqrt{t}} \circ \Phi^X_{-\sqrt{t}} \circ \Phi^Y_{\sqrt{t}} \circ \Phi^X_{\sqrt{t}})(x)</math>
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| ===In coordinates===
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| Though neither definition of the Lie bracket depends
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| on a choice of coordinates, in practice one often wants to compute
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| the bracket with respect to a coordinate system.
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| If we have picked a coordinate chart on ''M'' with local coordinate functions <math>\{x^i \}</math>, and we write <math>\partial_i = \frac{\partial}{\partial x^i}</math> for the associated local basis for the tangent bundle, then the vector fields can be written as
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| :<math>X=\sum_{i=1}^n X^i \partial_i</math>
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| and
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| :<math>Y=\sum_{i=1}^n Y^i \partial_i</math>
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| with smooth functions <math>X^i:M\to\mathbb{R}</math> and <math>Y^i:M\to\mathbb{R}</math>. Then the Lie bracket is given by
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| :<math>[X,Y] := \sum_{i=1}^n\left(X(Y^i) - Y(X^i)\right) \partial_i = \sum_{i=1}^n \sum_{j=1}^n \left(X^j \partial_j Y^i - Y^j \partial_j X^i \right) \partial_i </math>
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| If ''M'' is (an open subset of) '''R'''<sup>''n''</sup>, then the vector fields ''X'' and ''Y'' can be written as smooth maps of the form <math>X:M\to\mathbb{R}^n</math> and <math>Y:M\to\mathbb{R}^n</math>, and the Lie bracket <math>[X,Y]:M\to\mathbb{R}^n</math> is given by
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| :<math>[X,Y] := J_Y X - J_X Y</math>
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| where <math>J_Y</math> and <math>J_X</math> are the [[Jacobian matrix|Jacobian matrices]] of <math>Y</math> and <math>X</math>, respectively. These ''n''-by-''n'' matrices are multiplied by the ''n''-vectors ''X'' and ''Y''.
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| ==Properties==
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| The Lie bracket of vector fields equips the real vector space <math>V=\Gamma(TM)</math> of all vector fields on ''M'' (i.e., smooth sections of the tangent bundle <math>TM</math> of <math>M</math>) with the structure of a [[Lie algebra]], i.e., [·,·] is a map from <math>V\times V</math> to <math>V</math> with the following properties
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| *<math>[\cdot,\cdot]</math> is '''R'''-bilinear
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| *<math>[X,Y]=-[Y,X]\,</math>
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| *<math>[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.\,</math> This is the [[Jacobi identity]].
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| An immediate consequence of the second property is that <math>[X,X]=0</math> for any <math>X</math>.
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| Furthermore, there is a "[[product rule]]" for Lie brackets. Given a smooth real-valued function ''f'' defined on ''M'' and a vector field ''Y'' on ''M'', we have a new vector field ''fY'', defined by multiplying the vector ''Y<sub>x</sub>'' with the number ''f''(''x''), at each point ''x''∈''M''. The Lie bracket of ''X'' and ''fY'' is then given by
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| *<math> [X, fY] = X(f) Y + f [X,Y]</math>
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| (where on the right-hand side we multiply the function ''X''(''f'') with the vector field ''Y'', and the function ''f'' with the vector field [''X,Y'']).
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| This turns the vector fields with the Lie bracket into a [[Lie algebroid]].
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| We also have the following fact:
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| '''Theorem:'''
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| <math>[X,Y]=0\,</math> iff the flows of ''X'' and ''Y'' commute locally, i.e. iff for every ''x''∈''M'' and all sufficiently small real numbers ''s'', ''t'' we have <math>(\Phi^Y_t \Phi^X_s) (x) =(\Phi^X_{s}\, \Phi^Y_t)(x)</math>.
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| == Examples ==
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| For a [[Lie group]] ''G'', the corresponding [[Lie algebra]] is the tangent space at the identity, which can be identified with the left invariant vector fields on ''G''. The Lie bracket of the Lie algebra is then the Lie bracket of the left invariant vector fields, which is also left invariant.
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| 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:
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| :<math>[X,Y] = XY - YX</math>
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| where juxtaposition indicates matrix multiplication.
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| == Applications ==
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| The Jacobi–Lie bracket is essential to proving [[small-time local controllability]] (STLC) for driftless [[affine control systems]].
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| ==Generalizations==
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| As mentioned above, the [[Lie derivative]] can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to [[vector-valued differential form]]s) is the [[Frölicher–Nijenhuis bracket]].
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| ==References==
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| * {{springer|title=Lie bracket|id=p/l058550}}
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| * {{citation|last=Isaiah|first=Pantelis|title=Controlled parking [Ask the experts]|journal=IEEE Control Systems Magazine|year=2009|volume=29|issue=3|pages=17–21, 132|doi=10.1109/MCS.2009.932394}}
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| * {{citation
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| | last = Khalil
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| | first = H.K.
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| | authorlink = Hassan K. Khalil
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| | year = 2002
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| | edition = 3rd
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| | url = http://www.egr.msu.edu/~khalil/NonlinearSystems/
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| | isbn = 0-13-067389-7
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| | title = Nonlinear Systems
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| | publisher=[[Prentice Hall]]
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| | location = Upper Saddle River, NJ}}
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| * {{Citation|author=Kolář, I., Michor, P., and Slovák, J.|title=Natural operations in differential geometry|url=http://www.emis.de/monographs/KSM/index.html|publisher=Springer-Verlag|year=1993}} Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
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| * {{Citation|author=Lang, S.|title=Differential and Riemannian manifolds|publisher=Springer-Verlag|year=1995|isbn=978-0-387-94338-1}} For generalizations to infinite dimensions.
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| * {{Citation|author=Lewis, Andrew D.|url=http://penelope.mast.queensu.ca/math890-03/ps/math890.pdf|title=Notes on (Nonlinear) Control Theory}}
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| *{{Citation | last =Warner | first = Frank | title = Foundations of differentiable manifolds and Lie groups | origyear = 1971 | edition = | year = 1983 | publisher=Springer-Verlag | location = New York-Berlin | isbn = 0-387-90894-3 }}
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| [[Category:Bilinear operators]]
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| [[Category:Binary operations]]
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| [[Category:Differential topology]]
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| [[Category:Riemannian geometry]]
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