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Recently I am studying lagrangian mechanics where I came across the topic "principle of least action" which states that a system always takes the path of least action or when the action is minimum but I cannot understand why it should be true so can anyone give me the mathematical proof behind it and what is the original Idea behind it and again I want to understand what action actually is? In lagrangian mechanics it is defined as the total path integration of difference between Kinetic energy and potential energy, but what does it actually defining?

Qmechanic
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Ganesh Khadanga
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    We have to assume something. However, while $\delta S=0$ is a "bedrock" assumption in classical mechanics, it admits a quantum motivation. – J.G. Feb 12 '22 at 10:52
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    Possible duplicates: https://physics.stackexchange.com/q/15899/2451 , https://physics.stackexchange.com/q/9/2451 and links therein. – Qmechanic Feb 12 '22 at 12:06
  • maybe my answer here will ground you on how physics theories are built. https://physics.stackexchange.com/q/582922/ . One cannot prove principles, laws, postulates in the same way one cannot prove the axioms of a purely mathematical theory. In mathematics, an axiom can be turned into a provable theorem, but then the theorem turns into an axiom. The same is true for the physics "axioms" (principles, laws, postulates). example – anna v Feb 12 '22 at 12:17
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    in euclidean geometry the "parallel lines do not meet" can be proven through theorems, but then a theorem has to be accepted axiomatically. . So by construction of the physics theories principles , as of least action, are "true" . if proven using theorems, then a theorem has to take the place of the principle , as assumed "true", – anna v Feb 12 '22 at 14:00

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This question is something I didn't completely understand when I was doing Classical Mechanics, but it's mostly up to how it's taught. We begin with $\delta S=0$, then construct the Lagrangian s.t. it reproduces Newton's Second Law $\textbf{F}=m\textbf{a}$ (or more accurately $\textbf{F}=-\nabla U$ where $U$ is the potential energy. The variational principle is used as a tool to produce a new way to reproduce Newton's Second Law, which governs all of Classical Mechanics.

Redcrazyguy
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