We analyze the mechanism of spontaneous symmetry breaking of scale invariance in Galilean invariant field theories. We show that there is no dynamical gapless dilaton mode unless the $U(1)$ particle number symmetry is spontaneously broken. When both scale and particle number symmetries are spontaneously broken there is one propagating gapless Nambu-Goldstone mode whose nature depends crucially on the existence of a chemical potential. We discuss the reversibility of RG flows in such theories.