How does classical mechanics change if motion was not infinitely differentiable?

Many "theoretical mechanicians" seem to awesome that motion is a [tex]{C^\infty }[/tex] function(at least that is how I learned it). However, it seems like the postulates of Newtonian/Lagrangian/Hamiltonian/Vakonomic mechanics seem to "work" in the general case where only the motion is a [tex]{C^2}[/tex](ie the acceleration always exists, but the jerk does not).

My question is how classical mechanics would change if we assume the general case where the motion of a particle is only guaranteed to be twice differentiable? Are there any MAJOR changes?