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

[Pinu7]

Many "theoretical mechanicians" seem to awesome that motion is a $${C^\infty }$$ 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 $${C^2}$$(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?

 

[Andy Resnick]

Good question. My guess is 'no'. The Langevin equation (Brownian motion) results in perfectly reasonable physics.

 

[alan.b]

Are you talking about "Achilles and the tortoise" and Zeno's paradoxes?