  We develop the framework for testing Lorentz invariance in the dark matter
sector using galactic dynamics. We consider a Lorentz violating (LV) vector
field acting on the dark matter component of a satellite galaxy orbiting in a
host halo. We introduce a numerical model for the dynamics of satellites in a
galactic halo and for a galaxy in a rich cluster to explore observational
consequences of such an LV field. The orbital motion of a satellite excites a
time dependent LV force which greatly affects its internal dynamics. Our
analysis points out key observational signatures which serve as probes of LV
forces. These include modifications to the line of sight velocity dispersion,
mass profiles and shapes of satellites. With future data and a more detailed
modeling these signatures can be exploited to constrain a new region of the
parameter space describing the LV in the dark matter sector.
