  In this paper we deal with a general second order continuous dynamical system
associated to a convex minimization problem with a Fr\`echet differentiable
objective function. We show that inertial algorithms, such as Nesterov's
algorithm, can be obtained via the natural explicit discretization from our
dynamical system. Our dynamical system can be viewed as a perturbed version of
the heavy ball method with vanishing damping, however the perturbation is made
in the argument of the gradient of the objective function. This perturbation
seems to have a smoothing effect for the energy error and eliminates the
oscillations obtained for this error in the case of the heavy ball method with
vanishing damping, as some numerical experiments show. We prove that the value
of the objective function in a generated trajectory converges in order O(1/t^2)
to the global minimum of the objective function. Moreover, we obtain that a
trajectory generated by the dynamical system converges to a minimum point of
the objective function.
