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I'm tying to understand, historically, what lead to Lagrangian mechanics (LM). What did Lagrange actually do?

In the time (year 1788), when Lagrange published his work (that we nowadays call "Lagrangian mechanics") the term "energy" was not known. It slowly developed in the next 60 years after Lagrange's work. However, in modern times we say that $L = T - V$ is the difference between two energies, then do some derivation magic and get equations of motions for particles (or density fields).

Today, LM is also often introduced in a way that claims that Newton laws are not optimal because they are so hard to solve because Newton did not know about "generalized coordinates". But surely Isaac Newton was smart enough to express $x = r \sin(\phi)$ or something similar in his times already. So, this approach to introduce LM also only fits if you already know the result of LM. (And even if Newton really did not know to use other coordinates than $x$, $y$ and $z$: you still can do this, today, without using LM explicitly)

So, what did Lagrange actually do in his original works? What problem did he tried to solve? And what did he call his resulting variables, if not "energy"?

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  • $\begingroup$ en.wikipedia.org/wiki/Joseph-Louis_Lagrange has a variety of information as a starting point. The original works are, well, quite long, and he addressed many problems, so this is really really broad. $\endgroup$
    – Jon Custer
    May 19, 2022 at 15:53
  • $\begingroup$ As far as I understand, Lagrange's reworking of Newtonian mechanics was a purely mathematical theory totally detached from mechanical analogy. If I recall correctly, Lagrange's theory is what inspired Maxwell to make the final revision to his theory of electromagnetism by removing any mechanical analogy in order to present a purely mathematical theory. This purely mathematical approach continues to this day. $\endgroup$
    – nwr
    May 19, 2022 at 18:18
  • $\begingroup$ The comment by @nwr is extremely strange. Lagrange's book Mechanique Analytique is about statics and dynamics. Mechanics is the very subject of the book. As to Maxwell: Maxwell's mechanical model was an exploratory tool, it was never intended as an actual model. The mechanical model guided derivation of mathematical expressions. In the abstract these expressions have general validity; they are valid for any medium with the properties as observed in experiments. The proof that the mechanical model served its purpose: Maxwell derived the speed of electromagnetic waves. $\endgroup$
    – Cleonis
    May 26, 2022 at 8:58
  • $\begingroup$ @nwr Continuing about Maxwell: Maxwell knew all along that the implementation details of the mechanical model were certainly off. However, Maxwell worked in such a way that the structure of the mathematical framework had general validity, independent of the implementation details. Comparison: the structure of the mathematics of Caloric theory transferred to the kinetic theory of heat. Caloric theory gave rise to the concept of entropy. The expressions for entropy carried over to kinetic theory of heat. Caloric as a substance was abandoned; the mathematical structure transferred. $\endgroup$
    – Cleonis
    May 26, 2022 at 9:08
  • $\begingroup$ @Cleonis As far as I understand when Maxwell originally published his EM theory complete with mechanical analogy (hexagons and spheres...) it was met with considerable criticism. Poincaré, for example, stated that there was a "feeling of discomfort and even mistrust amongst his countrymen when faced with Maxwell's mechanism". It was only a couple of years later that JCM returned to the theory and revised to its final form. The biography I read of JCM by Clegg states that Lagrange's mechanics influenced JCM's choice. Regardless, you are probably correct. I'm only just learning a bit about it. $\endgroup$
    – nwr
    May 27, 2022 at 18:55

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I recently came across an article that helped me put Lagrange's work in a very interesting perspective:

Author: Craig Fraser (historian of science)

Subject: d'Alembert's Traité de Dynamique

http://homes.chass.utoronto.ca/~cfraser/Dalembert.pdf

http://homes.chass.utoronto.ca/~cfraser/D%27Alembert2.pdf

Craig Fraser mentions: in the time of Euler, d'Alembert, etc. the way that differential calculus was used was significantly different from the modern use. The application of differential calculus was very much tied up in geometric interpretation. As I understand it: wherever a theorem was stated and proved a diagram was an essential part of the statement/proof.

As I understand it: part of Lagrange's research programme was to reformulate mathematics so as to completely remove any dependence on diagrams. As I understand it: the title of his main work: "Mecanique Analytique" refers to that. All theorems are stated purely in the from of formulas, no dependence on geometric interpretation.


It's not that prior to Lagrange generalized coordinates were not used at all. Specific problems were solved with ingenious methods, that in retrospect can be recognized using a form of generalized coordinates. But that was on a case-by-case basis. My understanding is that Lagrange made usage of generalized coordinates systematic.

About energy:
We have that Lagrange used the energy concept systematically, and that he could show that this energy mechanics reproduces the newtonian formulation. Many authors write that to that end Lagrange used something they refer to as d'Alembert's principle. However, it could be that Lagrange attributed to d'Alembert something that was more Lagrange's interpretation.

There can be no doubt that Lagrange did have a demonstration of equivalence of newtonian formulation and energy mechanics. To my understanding: that demonstration did not use (at least not explicitly), the theorem that today is known as the Work-Energy theorem.

(There is a derivation of the Work-Energy theorem from F=ma by me on physics.stackexchange.)

A translation of 'Mecanique Analytique' is available on archive.org Analytic mechanics
(Among the download options there is an option to download as .pdf file.)

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