  We derive the geodesic equation of motion in the presence of weak
gravitational fields produced by relativistic sources such as cosmic strings,
decomposed into scalar, vector and tensor parts. We find that the vector
(gravito-magnetic) force is an important contributor, and for non-relativistic
particles we recover the well-known result for the impulse from a moving
straight string. Our results can be straightforwardly incorporated into N-body
simulations to allow for the presence of cosmic defects or other sources of
weak gravitational fields.
