  We study the Moyal evolution of the canonical position and momentum
variables. We compare it with the classical evolution and show that, contrary
to what is commonly found in the literature, the two dynamics do not coincide.
We prove that this divergence is quite general by studying Hamiltonians of the
form $p^2 /2m + V(q)$. Several alternative formulations of Moyal dynamics are
then suggested. We introduce the concept of starfunction and use it to
reformulate the Moyal equations in terms of a system of ordinary differential
equations on the noncommutative Moyal plane. We then use this formulation to
study the semiclassical expansion of Moyal trajectories, which is cast in terms
of a (order by order in $\hbar$) recursive hierarchy of i) first order partial
differential equations as well as ii) systems of first order ordinary
differential equations. The latter formulation is derived independently for
analytic Hamiltonians as well as for the more general case of smooth local
integrable ones. We present various examples illustrating these results.
