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

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SUMMARY

The discussion centers on the implications of assuming that motion is only twice differentiable ({C^2}) rather than infinitely differentiable ({C^\infty}). Participants assert that classical mechanics, including Newtonian, Lagrangian, Hamiltonian, and Vakonomic frameworks, remain valid under the {C^2} condition, as acceleration exists even if jerk does not. The Langevin equation, which describes Brownian motion, further supports the notion that classical mechanics can accommodate this limitation without major changes to its foundational principles.

PREREQUISITES
  • Understanding of classical mechanics principles, including Newtonian and Lagrangian mechanics.
  • Familiarity with mathematical concepts of differentiability, specifically {C^\infty} and {C^2} functions.
  • Knowledge of the Langevin equation and its application in describing Brownian motion.
  • Awareness of philosophical implications in mechanics, such as Zeno's paradoxes.
NEXT STEPS
  • Research the implications of {C^2} differentiability on classical mechanics.
  • Study the Langevin equation in detail to understand its role in classical and statistical mechanics.
  • Explore Zeno's paradoxes and their relevance to motion and differentiability in physics.
  • Investigate the differences between classical mechanics and modern physics frameworks under varying assumptions of motion.
USEFUL FOR

The discussion is beneficial for theoretical physicists, mathematicians, and students of mechanics who are exploring the foundations of motion and its mathematical representations.

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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