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This paper deals with the theoretical and numerical treatment of the unilateral dynamic contact problem between two arbitrary elastic bodies without friction. In addition to the classical variational statement that arises from static problems, a dynamic contact condition is needed and found by adjusting the balance laws of physical quantities to the impenetrability condition. In the context of infinitesimal deformation, a reciprocal formulation is then used to reduce this well-posed problem to one involving Green functions only defined on the contact surface. It is then often possible to approximate the system using considerably fewer unknowns than with finite difference algorithms. Analytical and numerical simulations are conducted and the results are reported and discussed.