The chemical rate network responsible for the formation of molecular hydrogen was incorporated into an N-body hydrodynamic code, in order to study the formation of the first cosmological objects at redshifts between 10 and 50. The implementation of the chemical and cooling processes was tested by comparing top hat simulations with theoretical predictions from a semi-analytic model and was in good agreement. One objective was to determine the minimum mass (MSG(z)) of perturbations that could become self-gravitating (a prerequisite for star formation), and the redshift at which this occurred. Cosmological simulations with realistic initial conditions produced primordial objects that became self-gravitating at redshifts in agreement with the MSG(z) results from the top hat simulations. The rotation of the core gas was directly related to its mass, indicating that greater mass was required to reach the self-gravitating state in the presence of rotational support. The universal fraction of self-gravitating gas increased from 10-4 at z = 23 to 5 x 10-3 at z = 10, and provides an upper constraint on the efficiency of early star formation.