Measurements of normalized differential cross sections of top quark pair (t (t) over bar) production are presented as a function of the mass, the transverse momentum and the rapidity of the t (t) over bar system in proton-proton collisions at center-of-mass energies of root s = 7 and 8 TeV. The data set corresponds to an integrated luminosity of 4.6 fb(-1) at 7 TeV and 20.2 fb(-1) at 8 TeV, recorded with the ATLAS detector at the Large Hadron Collider. Events with top quark pair signatures are selected in the dilepton final state, requiring exactly two charged leptons and at least two jets with at least one of the jets identified as likely to contain a b hadron. The measured distributions are corrected for detector effects and selection efficiency to cross sections at the parton level. The differential cross sections are compared with different Monte Carlo generators and theoretical calculations of t (t) over bar production. The results are consistent with the majority of predictions in a wide kinematic range.