  Non-perturbative constraints on many body physics--such as the famous
Lieb-Schultz-Mattis theorem--are valuable tools for studying strongly
correlated systems. To this end, we present a number of non-perturbative
results that constrain the low-energy physics of systems having conserved
dipole moments. We find that for these systems, a unique translationally
invariant gapped ground state is only possible if the polarization of the
system is integer. Furthermore, if a lattice system also has $U(1)$ subsystem
charge conservation symmetry, a unique gapped ground state is only possible if
the particle filling along these subsystems is integer. We also apply these
methods to spin systems, and determine criteria for the existence of a new type
of magnetic response plateau in the presence of a non-uniform magnetic field.
Finally, we formulate a version of Luttinger's theorem for 1D systems
consisting of dipoles.
