Long time molecular dynamics simulations of one-dimensional Lennard-Jones systems reveal that while the diffusion coefficient of a tagged particle indeed goes to zero in the very long time, the mean-square displacement is linear with time at short to intermediate times, allowing the definition of a short time diffusion coefficient [Lebowitz and Percus, Phys. Rev. 155, 122 (1967)]. The particle trajectories show intermittent displacements, surprisingly similar to the recent experimental results [Wei , Science 287, 625 (2000)]. A self-consistent mode coupling theory is presented which can partly explain the rich dynamical behavior of the velocity correlation function and also of the frequency dependent friction. The simulations show a strong dependence of the velocity correlation function on the size of the system, quite unique to one dimensional interacting systems. Inclusion of background noise leads to a dramatic change in the profile of the velocity time correlation function, in agreement with the predictions of Percus [Phys. Rev. A 9, 557 (1974)].