TITLE: Physical interpretation of the Lie Bracket
QUESTION [6 upvotes]: I've come accross this physical interpretation for $ [X,Y] $ which I don't understand :

Follow $X$ for some time $\epsilon$;
Follow $Y$ for $\epsilon$;
Follow -X for $\epsilon$;
Follow -Y for $\epsilon$;

In the limit as $\epsilon$ approaches 0, the result of the above motion approaches the Lie Bracket $[X,Y]$.
Maybe someone can elucidate this for me?

REPLY [2 votes]: The phrase 'flow $g$ along $Y$ a small distance $\epsilon$' is very nice, but all it means ultimately is just 'take the directional derivative of the function (e.g. surface) $g$ along $Y$'. Thus $X(Y)g$ means 'take the directional derivative of $Yg$ along $X$'. But $Yg$ is itself a directional derivative. Thus, $X(Y)g$ means 'take the directional derivative along $X$ of the directional derivative of $g$ along $Y$'.
Intuitively this is a generalisation of $\frac{\partial^2 g}{\partial x \partial y}$, since in the Lie bracket the two vector fields $X$ and $Y$ do not have to be orthogonal.
The second half of the Lie bracket then subtracts the same derivations in reverse order. If the two derivations commute, the Lie bracket is zero.
The vector flow terminology has definite aesthetic appeal (my original background is in fluid mechanics), but it remains difficult for me to visualise intuitively how the four arcs shrinking to a point as $\epsilon\rightarrow 0$ end up with this double derivation. The slope of a surface $g(x,y)$ in a particular direction at a given point $(x,y)$, on the other hand, is immediately obvious; and how that slope then may change in the direction of the other vector field is also intuitively clear. Once this has been established, the fact that these two double derivations may differ holds no mysteries.
Of course, the picture of the four arcs and whether they close or not remains the best visualisation of the Lie bracket once what it means has been understood.