I've run into a bit of a sticky point in doing some background research about the Hamiltonian structure of KdV.
What I know is:
A Poisson bracket is written in coordinates $x$ as $$\{F, G\} = (\nabla F)^T J(x) (\nabla G)$$
For some mapping $\phi: M \rightarrow M$, where $M$ is the manifold on which our dynamical system is defined, $\phi$ is a Poisson mapping if
$$\{F \circ \phi, G \circ \phi\} = \{F, G\}\circ \phi$$
This is all fine. But then, I have a source that says: The time map of a Hamiltonian dynamical system
$$\dot{x} = \{x, H\} = J(x)H(x)$$ is a Poisson map.
I don't understand what "the time map" is, or what it applies to. I see how the second equality above makes sense, and I see how KdV can be written as $\dot{u} = \{u, H(u)\}$, but I'm trying to make sense of what is being preserved exactly when I use a symplectic integrator, for example, or when I integrate in time.
Any clarification would be very helpful. Thanks.