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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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    In your case, you can notice that v.f. $V$ does not change the direction relative to the flow of v.f. $U$, and moreover the magnitude of $V$ changes at the same rate as the one of $U$, so the derivative of $V$ w.r.t. $U$ will not see any change at all, i.e. it will give 0.2012-04-17
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    More generally, one of the geometrical meanings of commutativity (zero bracket) of vector fields relies on the flows generated by these two vectors. Two vector fields are commutative if and only if its flows are too, in the sense that there is no difference starting at one point $p$, traveling a time $t_1$ over the flow of, lets say, $X_1$ and then a time $t_2$ over the flow of, say, $X_2$, or, instead, traveling first $t_2$ over the flow of $X_2$ and then $t_1$ over the flow of $X_1$. You end up at the same point if and only if these two vector fields have zero bracket.2012-04-22
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    Note that U is a dilation and V is a rotation. Therefore they commute.2012-08-27
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    @matgaio that is a wonderful answer! I can't believe I didn't know that before. Is that a common interpretation, or can you recommend a reference which uses it?2012-08-28
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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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