  In this work, we approach the minimization of a continuously differentiable
convex function under linear equality constraints by a second-order dynamical
system with an asymptotically vanishing damping term. The system under
consideration is a time rescaled version of another system previously found in
the literature. We show fast convergence of the primal-dual gap, the
feasibility measure, and the objective function value along the generated
trajectories. These convergence rates now depend on the rescaling parameter,
and thus can be improved by choosing said parameter appropriately. When the
objective function has a Lipschitz continuous gradient, we show that the
primal-dual trajectory asymptotically converges weakly to a primal-dual optimal
solution to the underlying minimization problem. We also exhibit improved rates
of convergence of the gradient along the primal trajectories and of the adjoint
of the corresponding linear operator along the dual trajectories. Even in the
unconstrained case, some trajectory convergence result seems to be new. We
illustrate the theoretical outcomes through numerical experiments.
