  We derive rigorously the reduced dynamical laws for quantized vortex dynamics
of the nonlinear Schr\"{o}dinger equation on the torus with non-vanishing
momentum when the vortex core size {\epsilon} \to 0. The reduced dynamical laws
are governed by a Hamiltonian flow driven by a renormalized energy. A key
ingredient is to construct a new canonical harmonic map to include the effect
from the non-vanishing momentum into the dynamics. Finally, some properties of
the reduced dynamical law are discussed.
