  We illustrate how the different kinds of constraints acting on an impulsive
mechanical system can be clearly described in the geometric setup given by the
configuration space--time bundle $\pi_t:\mathcal{M} \to \mathbb{E}$ and its
first jet extension $\pi: J_1 \to \mathcal{M}$ in a way that ensures total
compliance with axioms and invariance requirements of Classical Mechanics. We
specify the differences between geometric and constitutive characterizations of
a constraint. We point out the relevance of the role played by the concept of
frame of reference, underlining when the frame independence is mandatorily
required and when a choice of a frame is an inescapable need. The thorough
rationalization allows the introduction of unusual but meaningful kinds of
constraints, such as unilateral kinetic constraints or breakable constraints,
and of new theoretical aspects, such as the possible dependence of the
impulsive reaction by the active forces acting on the system.
