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.