Lie Bracket of Vector Fields
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High Quality Content by WIKIPEDIA articles! In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi Lie bracket, or Commutator of vector fields 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].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 ...
High Quality Content by WIKIPEDIA articles! In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi Lie bracket, or Commutator of vector fields 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].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.The Jacobi Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.
