Lie bracket of vector fields

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 that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y].

Conceptually, the Lie bracket [X, Y] is the derivative of Y along the flow generated by X, and is sometimes denoted ${\displaystyle {\mathcal {L}}_{X}Y}$ ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X.

The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.

The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems.^[1]

V. I. Arnold 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 float are flowing according to vector field X, and the fisherman lengthens/shrinks and turns the fishing rod according to vector field Y. The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.^[2]