  Vortex motion in type II superconductors is studied starting from a variant
of the time dependent Ginzburg-Landau equations, in which the order parameter
relaxation time is taken to be complex. Using a method due to Gor'kov and
Kopnin, we derive an equation of motion for a single vortex ($B\ll H_{c2}$) in
the presence of an applied transport current. The imaginary part of the
relaxation time and the normal state Hall effect both break ``particle-hole
symmetry,'' and produce a component of the vortex velocity parallel to the
transport current, and consequently a Hall field due to the vortex motion.
Various models for the relaxation time are considered, allowing for a
comparison to some phenomenological models of vortex motion in superconductors,
such as the Bardeen-Stephen and Nozi\`eres-Vinen models, as well as to models
of vortex motion in neutral superfluids. In addition, the transport energy,
Nernst effect, and thermopower are calculated for a single vortex. Vortex
bending and fluctuations can also be included within this description,
resulting in a Langevin equation description of the vortex motion. The Langevin
equation is used to discuss the propagation of helicon waves and the
diffusional motion of a vortex line. The results are discussed in light of the
rather puzzling sign change of the Hall effect which has been observed in the
mixed state of the high temperature superconductors.
