  In this paper, the theory of smooth action-dependent Lagrangian mechanics
(also known as contact Lagrangians) is extended to a non-smooth context
appropriate for collision problems. In particular, we develop a Herglotz
variational principle for non-smooth action-dependent Lagrangians which leads
to the preservation of energy and momentum at impacts. By defining
appropriately a Legendre transform, we can obtain the Hamilton equations of
motion for the corresponding non-smooth Hamiltonian system. We apply the result
to a billiard problem in the presence of dissipation.
