Externally Induced Dissipative Collisions

In this Chapter, we consider a problem which is related to the so-called standard inelastic shocks, in the general formulation given by Moreau in [Mor 11]. Again, we shall be dealing with a material point or a system (mechanical or otherwise) with finite number of degrees of freedom, which is represented by a point in an Euclidian space E. The scalar product in E is such that the kinetic energy for a motion q: I ⊂ R → E is given by \( \frac{1}{2}{\left\| {\dot q} \right\|^2} \). A continuous force field p : I × E → E, (t, q) → p(t, q), acts upon the system, which would obey to Lagrange’s equation of motion q(t) = p(t, q(t)) if it were free. Instead, we suppose that, by means of some external mechanism, the system is subjected to unilateral constraints which can produce collisions, i. e., shocks, and that these are dissipative, purely inelastic.