  A microscopic analysis of the non-dissipative force F_nd acting on a line
vortex in a type-II superconductor at T=0 is given. All work assumes a charged
BCS superconductor. We first examine the Berry phase induced in the BCS ground
state by movement of the vortex and show how this phase enters into the
hydrodynamic action S_hyd of the condensate. Appropriate variation of S_hyd
gives F_nd and variation of the Berry phase term is seen to contribute the
Magnus force of classical hydrodynamics to F_nd. This analysis confirms in
detail the arguments of Ao and Thouless within the context of the BCS model.
Our Berry phase, in the limit e -> 0, reproduces the Berry phase they obtain
for a neutral superfluid. A second independent determination of F_nd is given
through a microscopic derivation of the continuity equation for the condensate
linear momentum. This yields the acceleration equation for the superflow. The
vortex is seen to act as a momentum sink and the rate of momentum loss yields
F_nd. Both calculations yield the same F_nd and show that the Magnus force
contribution to F_nd is a consequence of the vortex motion and topology.
