Most of well-known approaches for rigid body simulations are formulated in the contact-space. Thanks to Gauss’ principle of least constraints, the frictionless dynamics problems are formulated in a motion-space. While the two formulations are mathematically equivalent, they are not computationally equivalent. The motion-space formulation is better conditionned, always sparse, needs less memory, and avoids some unnecessary computations. A preliminary experimental comparison suggests that an algorithm operating in the motionspace takes advantage of sparsity to perform increasingly better than a contact-space algorithm as the average number of contact points per object increases.