  We consider a general Langevin dynamics for the one-dimensional N-particle
Coulomb gas with confining potential $V$ at temperature $\beta$. These dynamics
describe for $\beta=2$ the time evolution of the eigenvalues of $N\times N$
random Hermitian matrices. The equilibrium partition function -- equal to the
normalization constant of the Laughlin wave function in fractional quantum Hall
effect -- is known to satisfy an infinite number of constraints called Virasoro
or loop constraints. We introduce here a dynamical generating function on the
space of random trajectories which satisfies a large class of constraints of
geometric origin. We focus in this article on a subclass induced by the
invariance under the Schr\"odinger-Virasoro algebra.
