It is postulated that all dynamical laws must be formulated in terms of variables measured relative to interacting particles. This implies that the two-particle system is the fundamental interaction unit. When this postulate is applied to Newtonian mechanics, a radically new conceptual structure is revealed. The two major changes which result are the absence of the law of inertia and a broadening of the space and time invariance to include accelerated observers. Although energy and angular momentum remain conserved in this reformulation, the usual linear momentum theorem does not appear. It is further shown that the theory can be expressed in the Lagrangian and Hamiltonian formalisms in a straightforward manner. Finally, applications are made to several simple two-body and three-body systems which demonstrate that many-particle interactions can be formulated without reference to the dynamics of a single particle.