  The initial value problem for the Ginzburg-Landau-Schr\"odinger equation is
examined in the $\epsilon \rightarrow 0$ limit under two main assumptions on
the initial data $\phi^\epsilon$. The first assumption is that $\phi^\epsilon$
exhibits $m$ distinct vortices of degree $\pm 1$; these are described as points
of concentration of the Jacobian $[J\phi^\epsilon]$ of $\phi^\epsilon$. Second,
we assume energy bounds consistent with vortices at the points of
concentration. Under these assumptions, we identify ``vortex structures'' in
the $\epsilon \rightarrow 0$ limit of $\phi^\epsilon$ and show that these
structures persist in the solution $u^\epsilon(t)$ of $GLS_\epsilon$. We derive
ordinary differential equations which govern the motion of the vortices in the
$\epsilon \rightarrow 0$ limit. The limiting system of ordinary differential
equations is a Hamitonian flow governed by the renormalized energy of Bethuel,
Brezis and H\'elein. Our arguments rely on results about the structural
stability of vortices which are proved in a separate paper.
