Geometric intuition behind the Lie bracket of vector fields

Question

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 $[U, V] = 0$ ?

Answer 1

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.

Answer 2

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.