  We find a relationship between the dynamics of the Gaussian wave packet and
the dynamics of the corresponding Gaussian Wigner function from the
Hamiltonian/symplectic point of view. The main result states that the momentum
map corresponding to the natural action of the symplectic group on the Siegel
upper half space yields the covariance matrix of the corresponding Gaussian
Wigner function. This fact, combined with Kostant's coadjoint orbit covering
theorem, establishes a symplectic/Poisson-geometric connection between the two
dynamics. The Hamiltonian formulation naturally gives rise to corrections to
the potential terms in the dynamics of both the wave packet and the Wigner
function, thereby resulting in slightly different sets of equations from the
conventional classical ones. We numerically investigate the effect of the
correction term and demonstrate that it improves the accuracy of the dynamics
as an approximation to the dynamics of expectation values of observables.
