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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 $U = x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$ and $V = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$, should it be obvious that $[U, V] = 0$ ?

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    @AdamSaltz, this is a kind of common interpretation and proofs for this can be found in several textbooks on analysis on manifolds, for example Lee's "Smooth Manifolds" (a great book!). What amuses me on this is the fact that it is a really natural question to know when two vector fields commute (in the sense I gave before) and that the answer is a simple algebraic calculation.2012-09-02

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

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    @sifsa See https://drive.google.com/folderview?id=0B6x6GQ82vuH_YnE2STNKZXU1cUk page 19 at the bottom. The proof exploits a definition of the Lie bracket by its effect on a test function $\varphi$ via $L_v(L_w \varphi) - L_w(L_v \varphi) = L_{[v,w]}\varphi$. Then it is sufficient to know that $L_v \varphi$ transforms canonically...2016-03-29
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matigaio gave the geometric meaning. Guessing if it vanishes or not require some practice by doing computations and drawing the vector fields (locally). You can for instance after drawing, try to see if it could commute or not by "following" the "quadrilateral" (sometimes open) made by integrating (Euler method, for small time) one vector field after the other one, etc.