History of Hamilton's principle

In summary, Hamilton determined that action, rather than any other quantity, should be stationary for a mechanical system by generalizing Newton's laws of motion to arbitrary coordinates. This was based on the principle that virtual displacements from the true path of a particle should do no work, meaning that infinitesimally different paths should have similar energies. Hamilton applied this concept in an abstract world, relating variations in the path to the energies put on the system.
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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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1. What is Hamilton's principle?

Hamilton's principle, also known as the principle of stationary action, is a fundamental principle in classical mechanics that states that the motion of a physical system is determined by a single quantity, known as the action, which is minimized along the path of motion.

2. Who is Hamilton and why is this principle named after him?

William Rowan Hamilton was an Irish mathematician and physicist who is credited with formulating Hamilton's principle in 1834. This principle was named after him to honor his contributions to the field of classical mechanics.

3. How is Hamilton's principle different from Newton's laws of motion?

Newton's laws of motion describe the behavior of a physical system based on the forces acting upon it. In contrast, Hamilton's principle takes into account the entire path of motion and considers the concept of action, which is a combination of a system's kinetic and potential energies.

4. What are the applications of Hamilton's principle?

Hamilton's principle has numerous applications in physics, engineering, and other fields. It is commonly used in the study of classical mechanics, fluid dynamics, and electromagnetism, and has also been applied in quantum mechanics and relativity.

5. Can Hamilton's principle be extended to other branches of science?

While Hamilton's principle was initially formulated for classical mechanics, it has been extended to other branches of science, such as quantum mechanics and relativity, through the use of variational principles. It has also been used in the study of complex systems and optimization problems in economics and biology.

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