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

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Classical mechanics traditionally assumes motion is infinitely differentiable, but it can still function under the condition that motion is only twice differentiable. The fundamental principles of Newtonian, Lagrangian, Hamiltonian, and Vakonomic mechanics remain applicable even when acceleration exists but jerk does not. The discussion raises the question of whether major changes would occur in classical mechanics under this assumption, with some suggesting that significant alterations are unlikely. The Langevin equation, which describes Brownian motion, supports the idea that reasonable physics can still emerge from less stringent differentiability conditions. Overall, the consensus leans towards minimal impact on classical mechanics from this change in assumptions.
Pinu7
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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?
 
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Good question. My guess is 'no'. The Langevin equation (Brownian motion) results in perfectly reasonable physics.
 
Are you talking about "Achilles and the tortoise" and Zeno's paradoxes?
 

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