Defining Lie bracket intuitively looking at commuting flows I'm reading "Control Theory from the Geometric Viewpoint" of Agrachev. He comments:
"It is natural to suggest that a lower-order term in
the Taylor expansion of $(1.12)$ at $t = s = 0$ is responsible for commuting
properties of flows of the vector fields VI, V2 at the point q."
Why is this natural? And why is it clear that non-mixed first and second order derivatives are useless? I wrote taylor's expansion and it wasn't clear to me:
$$\gamma(t,s)= \gamma(0,0)+V_2(q)s+\frac{\partial^2 \gamma }{\partial s \partial t}(0,0)ts+ V_2(P_2(q))s$$
 A: As for "it is natural to suggest", you may read this as "I suggest". Or, you may read this as "if it IS explained by lower order terms, it would be terrible to overlook that". Alternatively, you might think about the many other situations you know in calculus where a geometric quantity is explained by lower order derivatives: velocity; acceleration; curvature; etc.
But, the best reason for it to be "natural" for lower order terms to explain commutativity is to actually understand why lower order terms explain commutativity.
And as for that, the expression $V_2(q)$ for the value of $\frac{\partial\gamma}{\partial s}$ at $t=s=0$ is independent of $V_1$, so how can that tell you anything about the interaction between $V_1$ and $V_2$? Similarly, the expression $\frac{\partial}{\partial s}|_{s=0} V_2(P_2^s(q))$ is also independent of $V_1$, so it is useless for the same reason.
Thus, out of all of the first and second order terms, the only one which depends on both $V_1$ and $V_2$ is the second mixed partial.
A: If you want to learn the intuition behind the differential geometry concepts, I suggest you look at "Applied Differential Geometry" by William Burke.
Many people dislike the book because it's too intuitive. They are mistaken, in my opinion. Read it and tell for yourself!
