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 . It is the specialization of the Lie derivative to the case of Lie differentiation of a vector field. Indeed, equals the Lie derivative .
It plays an important role in differential geometry and differential topology, and is also fundamental in the geometric theory for nonlinear control systems (Isaiah 2009, pp. 20–21, nonholonomic systems; Khalil 2002, pp. 523–530, feedback linearization).
A generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.
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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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