Let's take a constructive approach to get a better idea of how the
spaces (manifolds) and Hamiltonians $H$ relate.
The simplest space would be a 2D plane $(q,p)$, where $q$ denotes
position and $p$ denotes momentum. Ignoring constants, this is the
phase space of a free point particle traveling on a line $H(q,p)=p^2$,
and the harmonic oscillator $H(q,p)= q^2 + p^2$.
In these simple contexts, what does $dp \wedge dq = 0$ mean?
A more interesting question might be: why is the symplectic Euler
discretization stable? And why is it slighty skewed?