The first time I was introduced to the covariant derivative I didn't even realise that was another "kind" of derivative. Following Hamilton's principle taking an action such that: $$ S=\int g_{\mu\nu} \frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}d\tau . $$ If you find the extremes of this integral or just apply lagrange equations you actually end up getting the geodesic equation. My question is: was the covariant derivative kind of discovered this way or is this just a coincidence or how is it even possible that applying lagrange equations which are on the realm of classical mechanics we end up getting a covariant derivative out of nowhere?
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1$\begingroup$ Would History of Science and Mathematics be a better home for this question? $\endgroup$– Qmechanic ♦Commented Jan 29 at 13:48
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$\begingroup$ @Qmechanic just trying to get some insight on how classical mechanics appear to "magically" lead to covariant derivatives. So i would say it's more a physics question rather than a historical one. $\endgroup$– ÁlvaroCommented Jan 29 at 13:50
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Covariant derivatives were introduced by Ricci and Levi-Civita around 1901, following previous works by Christoffel. Their main goal was to find a derivation of tensors that was invariant under a change of coordinates (you can find the manuscript online, it is pretty clear for a modern eye and easy to read).
Ricci and Levi-Civita realized afterward in the same book that this covariant derivative can be linked to geodesics and hence to the minimization principle you are referring to. The reason why applying Euler-Lagrange (from classical mechanics) give you this derivative is because this derivative tells you how to "parallel transport" a vector (see Levi-Civita's paper in 1917) and this is exactly what the velocity of a path which extremalize the length (a geodesic) does.
