  The Gross-Pitaevskii (GP) or nonlinear Schr\"odinger equation relevant to
ultracold atomic gaseous Bose-Einstein condensates possess a modified
Schr\"odinger symmetry in two spatial dimensions, in the presence of a harmonic
trapping potential, an (artificial) constant magnetic field (or rotation) and
an electric field of a quadratic electrostatic potential. We find that a
variance and a center of a trapped gas with or without a vorticity can be
regarded as massive Nambu-Goldstone (NG) modes associated with spontaneous
breaking of the modified Schr\"odinger symmetry. We show that the Noether
theorem for the modified Schr\"odinger symmetry gives universal equations of
motion which describe exact time-evolutions of the trapped gases such as a
harmonic oscillation, a cyclotron motion and a breathing oscillation with
frequencies determined by the symmetry independently of the details of the
system. We further construct an exact effective action for all the NG modes.
