Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates.

But is there a way to “guess” what is the Lie bracket of two vector fields ? What is the geometric intuition ?

For instance, if we take U=xx+yy and V=yx+xy, should it be obvious that [U,V]=0 ?

Answer

One way to get a geometric intuition for the Lie bracket is to note Φ([U,V])=[Φ(U),Φ(V)], i.e. the Lie bracket transforms canonically under the diffeomorphism Φ. Now if we have a straightening ΦU of the vector field U (such that ΦU(U) is constant in our coordinate system), then [U,V] is just the derivative of V along (the constant direction) U in that coordinate system.

Attribution
Source : Link , Question Author : pokraka , Answer Author : Thomas Klimpel

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