History of Hamilton's principle

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SUMMARY

Hamilton's principle establishes that action is the stationary quantity for mechanical systems, a concept derived from his efforts to generalize Newton's laws of motion to arbitrary coordinates. He posited that virtual displacements from the true path of a particle should not result in significant energy differences, emphasizing that any deviation should not interfere with the particle's energy. This principle highlights the relationship between path variations and the energies acting on the system, fundamentally linking Hamiltonian mechanics to classical mechanics.

PREREQUISITES
  • Understanding of Newton's laws of motion
  • Familiarity with the concept of virtual displacements
  • Basic knowledge of mechanical systems and energy principles
  • Introduction to Hamiltonian mechanics
NEXT STEPS
  • Study Hamiltonian mechanics and its applications in physics
  • Explore the derivation of action as a stationary quantity in classical mechanics
  • Learn about the implications of virtual displacements in mechanical systems
  • Investigate the relationship between energy and path variations in physics
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Physics students, mechanical engineers, and researchers interested in classical mechanics and Hamiltonian dynamics will benefit from this discussion.

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How did Hamilton work out that action is the stationary quantity for a mechanical system?

I've seen proofs that action is stationary, but it's unclear to me how Hamilton worked out that action as opposed to some other quantity should be stationary.
 
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he attempted to generalise the Newton's laws of motion to arbitrary coordinates. bearing in mind that virtual displacements from the true path of the particle should do no work. i.e. if you throw a stone down a tower, and it takes route A, some other route B infinitesimally close to route A should not be very much different in energy to route A, and in fact changing the particles route should not interfere with the energies of the particle, and can be regarded as an External energy.
right so if you throw a tennis ball straight, you expect it to go straight, if it doesn't then there is some other force acting on it that changes its path. which basically Newton's law. Hamilton wrote this down in an abstract world, where he sought the variations of the path to be related to the energies put upon the system.
 
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