The Cayley variational principle for continuous-impact problems: a continuum mechanics based version in the presence of a singular surface

In 1857, Arthur Cayley presented a variational principle for a class of dynamical problems, which he designated as continuous-impact problems. Cayley exemplified this class by means of a chain hanging over the edge of a table and being set into motion by its own weight. In the following, we present a continuum mechanics based version of the Cayley principle. The moving portion of the chain mentioned by Cayley represents a variable-mass system, and he assumed that the particles of the chain experience a jump in their velocity when being taken into connection with the moving part. We accordingly study a body containing a non-material region of transition, within which certain entities suffer considerable changes of their spatial distribution, and which we replace by equivalent surface growth terms at a singular surface, in order to derive our version of the principle. The falling chain mentioned by Cayley is used as an example problem.