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Most of well-known approaches for rigid body simulations are formulated in the contact-space. Due 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 conditioned, always sparse, needs less memory, and avoids some unnecessary computations. A preliminary experimental comparison suggests that an algorithm operating in the motion-space takes advantage of sparsity to perform increasingly better than a contact-space algorithm as the average number of contact points per object increases.