How do we know the principle of least action is true?

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    Least action Principle
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Discussion Overview

The discussion centers around the principle of least action, exploring its validity, historical context, and the motivations behind its formulation. Participants express curiosity about its origins and how it relates to established physical laws and theories.

Discussion Character

  • Exploratory
  • Historical
  • Conceptual clarification

Main Points Raised

  • One participant questions the straightforwardness of the principle's truth and its origins, suggesting that Lagrange must have derived it from prior knowledge.
  • Another participant asserts that the principle gives the right answers, implying that its correctness is validated by its outcomes.
  • A participant expresses interest in the discovery of the principle, indicating a desire to understand its historical development.
  • It is noted that the principle can be seen as a mathematical abstraction of existing knowledge, such as the shortest distance between two points and Newton's laws, with connections to newer theories like quantum mechanics.
  • One participant mentions a historical perspective, suggesting that Lagrange's motivation might be linked to theological arguments, although details are lacking.
  • References to various resources, including books and articles, are provided for those seeking deeper understanding of the principle and related concepts like Hamilton's and D'Alembert's principles.

Areas of Agreement / Disagreement

Participants express varying degrees of curiosity and perspectives on the principle's origins and implications, but there is no consensus on its foundational truth or the specifics of its discovery.

Contextual Notes

Some discussions reference historical motivations and connections to other principles, but these are not fully explored or resolved within the thread.

Terilien
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How do we know that it is true? Most people don't give me a very straightforward answer, but I don't like that. Lagrange must have gotten it from somwhere.
 
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It gives the right answers. What more do you expect?
 
How Was It Discovered!?
 
In some sense, it's a mathematical abstraction of things we already knew [like "shortest distance between two points is a line", geometric optics, Newton's laws of motion]... and we take pleasure to find that newer theories [developed after (say) Lagrange] seem to fit into this scheme... for example, quantum mechanics.

Implicit in this discussion is how one finds action that yield the equations of motion.

If you are interested in Lagrange's motivation, that's more of a historical question... which might be answered in (say) https://www.amazon.com/dp/0486650677/?tag=pfamazon01-20.

You might find some starting points here in this wikipedia article
http://en.wikipedia.org/wiki/History_of_variational_principles_in_physics (standard caveats apply).

You might find this section from http://mitpress.mit.edu/SICM/ interesting
http://mitpress.mit.edu/SICM/book-Z-H-8.html#%_sec_1.1If you want something deeper, then you might have to look at (say) Arnold or Abraham-Marsden.
 
Last edited by a moderator:
Terilien said:
How Was It Discovered!?

Back when I was an undergrad, my analytical mechanics TA briefly mentioned that the principle of least action arose from some sort of theological argument. Unfortunately I don't have any further details on that.
 
another good book to learn these stuff is the "classical" Classical Mechanics book by Goldstein.

to understand these, need to get your mind around Hamilton's or D'Alembert principle
 

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