Non-smooth mechanics is a modeling approach in mechanics which does not require the time evolutions of the positions and of the velocities to be smooth functions.
[1] Due to possible impacts, the velocities of the mechanical system are allowed to undergo jumps at certain time instants in order to fulfill the kinematical restrictions.
Consider for example a rigid model of a ball which falls on the ground.
At the impact time instant, the velocity must jump to a post-impact velocity which is at least zero, or else penetration would occur.
Non-smooth mechanical models are often used in contact dynamics.