Definition
Each vector field X on a smooth manifold M may be regarded as a differential operator acting on smooth functions on M. Indeed, each vector field X becomes a derivation on the smooth functions C∞(M) when we define X(f) to be the element of C∞(M) whose value at a point p is the directional derivative of f at p in the direction X(p).
The space of derivations of C∞(M) is a Lie algebra under the operation . This Lie algebra structure can be transferred to the set of vector fields on M as follows.
The Jacobi–Lie bracket or simply Lie bracket, of two vector fields X and Y is the vector field such that
Such a vector field exists because the right hand side is a derivation of C∞(M), and the vector space of such derivations is known to be isomorphic to the space of vector fields on M by the map sending a vector field X to the derivation .
To make the connection to the Lie derivative, let be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field . The differential of each diffeomorphism maps the vector field Y to a new vector field . To pull-back the vector field one applies the differential of the inverse, . The Lie bracket is defined by
In particular, is the Lie derivative of the vector field with respect to . Conceptually, the Lie bracket is the derivative of in the `direction' generated by .
Though neither definition of the Lie bracket depends on a choice of coordinates, in practice one often wants to compute the bracket with respect to a coordinate system. Let be a set of local coordinate functions, and let denote the associated local frame. Then
(Here we use the Einstein summation convention)
Read more about this topic: Lie Bracket Of Vector Fields
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