Ideal characterizations of multiple impacts: A frame–independent approach by means of jet–bundle geometry

We present, in the geometric setup given by the space-time bundle $ \mathcal {M}\/$ and its first jet-extension $ J_1\/(\mathcal {M}\/)$, an ideal constitutive characterization based on the conservation of kinetic energy for a general mechanical system with a finite number of degrees of freedom in contact/impact with a multiple unilateral constraint $ \mathcal {C}\/$ comprising a finite number of regular constraints of codimension $ 1$. We prove that the geometric structures associated to the elements of $ \mathcal {C}\/$ determine a natural criterion to choose the simplest non-trivial constitutive characterization among the various possibilities preserving the kinetic energy in multiple impacts. We put this specific choice at the core of an algorithm that determines the right-velocity of the system once the massive properties of the system, the elements of the multiple constraint and the left-velocity of the system are known, in cases of both single and multiple contact/impact. We show the application of the algorithm in three significant examples: the Newton Cradle, the simultaneous impact of a disk with two disks at rest and in contact, the impact of a disk with a disk at rest and in contact with two other disks.