  The classical and quantum worldsheet theory describing nonrelativistic open
string theory in an arbitrary nonrelativistic open and closed string background
is constructed. We show that the low energy dynamics of open strings ending on
n coincident D-branes in flat spacetime is described by a Galilean invariant
U(n) Yang-Mills theory. We also study nonrelativistic open string excitations
with winding number and demonstrate that their dynamics can be encoded into a
local gauge theory in one higher dimension. By demanding conformal invariance
of the boundary couplings, the nonlinear equations of motion that govern the
consistent open string backgrounds coupled to an arbitrary closed background
(described by a string Newton-Cartan geometry, Kalb-Ramond, and dilaton field)
are derived and shown to emerge from a Galilean invariant Dirac-Born-Infeld
type action.
