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I was thinking about the foundations of geometric mechanics and its precursors. I wondered who was the first to realized the equivalence between Riemannian geometry and Lagrangian mechanics. In particular:

  1. Solutions (trajectories) of Lagrange equations are geodesics of Levi-Civita connection.
  2. The moment of inertia tensor can be viewed as a Riemannian metric tensor.

I would be grateful for any answers and relevant references to this topic.

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    $\begingroup$ I’m voting to close this question because it's better suited to hsm.stackexchange than here. $\endgroup$
    – user44191
    Commented May 27, 2020 at 0:03
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    $\begingroup$ Looks like a perfectly reasonable question to me, offering someone the opportunity to explain some interesting mathematics. Why is there no "vote against" on the closure link? $\endgroup$
    – Paul Taylor
    Commented May 28, 2020 at 11:57
  • $\begingroup$ (Anonymity? Man up.) There are knowledgble users of MO who refuse to use HSM, so better to post both places with a cross-ref. $\endgroup$
    – Tom Copeland
    Commented May 29, 2020 at 6:08

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Darboux was perhaps the first to argue fully explicitly that “the general problem of mechanics is nothing but the generalisation to an arbitrary number of variables of the problem of the study of geodesics” (Leçons sur la théorie générale des surfaces, Paris, 1889, p. 500). A very useful article on the (pre)history of this is Lützen, Interactions between mechanics and differential geometry in the 19th century.

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