I was reading this very nice document, to understand why symplectic geometry is the natural setting for classical mechanics. I more or less understood why there is naturally a 2-form that arises. However, I didn't really understand the argument towards the end :
[...]All that remains is to explain why ω should be closed, i.e., why dω = 0. Unfortunately I don't see how to explain that without slightly more notation, but it won't be too bad. This requirement corresponds to a slightly more subtle issue, namely that the laws of physics should not depend on time. Let Ft denote the time-t flow along the vector field corresponding to some Hamiltonian H. One natural way to get the laws of physics at time t is to look at the pullback Ft*ω. We want it to equal ω. Clearly it does so at time 0, so to check that it always does we simply differentiate with respect to t, as follows.
What does he mean by : "One natural way to get the laws of physics at time t is to look at the pullback $F_t^*\omega$"? Why is that? And why do we want it to be equal to $\omega$?