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Ragnar
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Could someone explain why the principle of least action is true?
Crosson said:Before learning this principle, you should have a good foundation with Newton's 2nd law.
Can you explain why the second law is true?
I suppose. Yes I can.
what action is and why it is minimized
Mentz114 said:We know nature hates potential energy and all motion is just potential energy
converting to kinetic or heat energy. But the transfer between the energies is alway subject to least action.
Crosson said:As I promised, action is the quantity which when minimized reproduces the same motions as would Newton's 2nd law. Similarly, the lagrangian is the thing which when substituted into lagrange's equations reproduces Newton's law.
Mentz is referring to the fact that any system will always seek to minimize its potential energy. This is why caculating the stability of a system only requires calculating the 2nd derivative of its potential energy, ie: its concavity at a particular point. If the system can shed additinal potential energy the it will do so until it reaches a local or absolute minimum.pardesi said:i can't understand what u mean when u say nature hates p.e??
This post is on Hamilton's Principle, not on Fermat's Principle. Both require the calculus of variation to derive a useful result. In the context of this post it is Langrange's equations, whereas Fermat's Principle leads to Snell's Law. However, I'm not so sure I find Fermat's Principle that intuitive either.pardesi said:as far as fermat's principle of least time goes it's a beautiful and intutive observation .
The Principle of Least Action is a fundamental principle in physics that states that a physical system will always follow the path that minimizes the action, or the integral of the system's Lagrangian over time. In other words, a system will always take the path of least resistance.
The Principle of Least Action is used in various fields of physics, such as classical mechanics, quantum mechanics, and electromagnetism. It is also used in other areas, such as economics and biology, to describe the behavior of complex systems.
The Principle of Least Action is derived from the Lagrangian formulation of classical mechanics, which uses the concept of virtual work to determine the equations of motion for a system. By setting the variation of the action to zero, the equations of motion can be derived, leading to the principle of least action.
The Principle of Least Action allows us to determine the path that a physical system will take by minimizing the action, which is a measure of the system's energy. This principle has been proven to accurately describe the behavior of a wide range of physical systems and is a powerful tool in understanding the laws of nature.
The Principle of Least Action has been used to explain the motion of celestial bodies in the solar system, the behavior of particles in quantum mechanics, and the path of light in optics. It has also been applied in engineering and economics to optimize systems and minimize energy consumption.