How to derive the 4 laws of motion using Calculus?

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SUMMARY

The discussion clarifies that the laws of motion, specifically Newton's laws, cannot be derived solely using calculus as they are fundamentally physical laws established through observation and experimentation. It highlights that while calculus can express these laws mathematically, such as in Newton's second law (F = ma), the laws themselves must be postulated. The conversation also notes the common confusion regarding the number of laws, emphasizing that there are three primary laws of motion, with the work-energy theorem and angular momentum theorem being derivations related to these laws.

PREREQUISITES
  • Understanding of Newton's three laws of motion
  • Basic knowledge of calculus, specifically derivatives
  • Familiarity with the work-energy theorem
  • Concept of angular momentum in physics
NEXT STEPS
  • Study the derivation of the work-energy theorem from Newton's second law
  • Explore the concept of angular momentum and its relation to motion
  • Review the historical context and formulation of Newton's laws of motion
  • Investigate the application of calculus in physics beyond classical mechanics
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Students of physics, educators teaching classical mechanics, and anyone interested in the mathematical foundations of physical laws.

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How do you derive the laws of motion using Calculus from scratch
 
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You can't. The laws of motion are physical laws, independent of the foundations of mathematics. The laws can be stated in mathematical form and developed using mathematics, such as calculus.

But, they have to be postulated through observation and experiment.
 
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First of all, I thought there were 3 Newton's laws of motion.

Second of all (ironically speaking), the Newton's 2nd law states that F = ma. We can say that a is equivalent to the derivative of velocity wrt time. The derivative, in this case, uses calculus.
 
I do not understand which 4 laws are supposed to derive but the work energy theorem is usually derived from the 2nd Newton; as well as the angular momentum theorem is also a theorem and it is derived.
 

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