$(M,w)$ is symplectic manifold. $f_t : M\to M$ is a symplectic isotopy between $f_0=id$ and $f_1$. Let X_t be the vector field on M satisfying $d(f_t)/dt=X_t(f_t)$ Now I differentiate $(f_t)^*w$. Here goes the problem. If $X_t$ is time(t) independent, I know that the result is the Lie derivative of w with respect to the vector field $X(write L(X)w)$, simply by definition. But for general $X_t$, Why is the result same as $L(X_t)w$?
The question arose while reading "Dusa McDuff, Dietmar Salamon Introduction to symplectic topology 1999". It appears at p83-84, proof of propostion 3.2. I also found someone's note on the internet but neither the book nor the note explains this. So I suspect that this is trivial problem but still not figured it out. Thanks for the help.
http://www.math.jussieu.fr/~ma/M2_14/Chap1.2-1.3.pdf
P.27-28, it's the exercise 1.2.26.
It's not trivial, but really makable, as you'll see.