  It is well known that compactifying a space can break symmetries that are
present in the covering space. In this paper we study the effects of such
topological symmetry breaking on point-particle motion when the particle is
coupled to a massless field on the space. For a torus topology where Lorentz
invariance is broken but translation invariance is maintained, particles can
move at a constant velocity through the space; however, non-local,
velocity-dependent forces arise whenever the particle is accelerated. For a
topology where translation invariance is broken, such as the Klein bottle,
interactions with the massless field generate an effective potential as a
function of position. The potential creates special stable points in the space,
and prevents constant velocity motion. This latter would appear to be the
generic case. This class of effects may be applicable whenever a localized
object moves through a compactified bulk, such as in brane-world cosmology, or
some condensed matter systems.
