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We construct a new geometric framework based on the concepts of left and right jet-bundles of a classical space-time V in order to analyze the impulsive behavior of a unilateral constraint S. The setup allows deep insights into how one can choose an ideality criterion for the constraint S when the hypothesis of conservation of kinetic energy is assumed. We show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of S is 1, and that it leaves the impulsive reaction partially undetermined when the codimension of S is greater than 1. If the codimension of S is greater than 1, we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of S. We show that both the Newton-like and the Poisson-like approaches to the description of the reactive impulse are equivalent, in the sense that both give the same results about the ideality criterion. Moreover, we prove that the same results hold using the classical approach based on reflection operators, possible only in case of codimension 1. We present also several physically meaningful examples. [ABSTRACT FROM AUTHOR]