Lie bracket of vector fields

In the mathematical field of <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, 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 <a href="/facts/Vector_field/ccFNuPPp">vector fields</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 on a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 a third vector field denoted 
 
 
 
 [
 X
 ,
 Y
 ]
 
 
 {\displaystyle [X,Y]}
 
.
Conceptually, the Lie bracket 
 
 
 
 [
 X
 ,
 Y
 ]
 
 
 {\displaystyle [X,Y]}
 
 is the derivative of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 along the <a href="/facts/Vector_flow/Dgk6RcrP">flow</a> generated by 
 
 
 
 X
 
 
 {\displaystyle X}
 
, and is sometimes denoted 
 
 
 
 
 
 
 L
 
 
 
 X
 
 
 Y
 
 
 {\displaystyle {\mathcal {L}}_{X}Y}
 
 ("Lie derivative of Y along X"). This generalizes to the <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> of any <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a> along the flow generated by 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
The Lie bracket is an R-<a href="/facts/Bilinear_operator/LH1lHVeP">bilinear</a> operation and turns the set of all <a href="/facts/Smoothness/mffL4ch2">smooth</a> vector fields on the manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 into an (infinite-dimensional) <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>.
The Lie bracket plays an important role in <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> and <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, for instance in the <a href="/facts/Frobenius_theorem_(differential_topology)/A8gu0fjt">Frobenius integrability theorem</a>, and is also fundamental in the geometric theory of <a href="/facts/Nonlinear_control_theory/KScKca1A">nonlinear control systems</a>.
<a href="/facts/Vladimir_Arnold/IMBhyfMS">V. I. Arnold</a> refers to this as the "fisherman derivative", as one can imagine being a fisherman, holding a fishing rod, sitting in a boat. Both the boat and the <a href="/facts/Fishing_float/rCqLXVwC">float</a> are flowing according to vector field 
 
 
 
 X
 
 
 {\displaystyle X}
 
, and the fisherman lengthens/shrinks and turns the fishing rod according to vector field 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.

Lie bracket of vector fields open-in-new

Lie bracket of vector fields