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The variational principle is named after Hamilton, instead of Lagrange.

So it seems that he did not derive his equation by the variational method.

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    $\begingroup$ Possible duplicate of who was the first to discover the Hamilton principle of classical physics? $\endgroup$ – Francois Ziegler Apr 13 '18 at 3:05
  • $\begingroup$ @FrancoisZiegler This question appears to be a "how" question, not a "who" question. $\endgroup$ – Geremia Apr 13 '18 at 3:11
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    $\begingroup$ @Geremia The other question asks “what exactly, was Lagrange’s role?”, and this is thoroughly answered in the quoted paper of Fraser (1983, p. 217); also (1985, p. 173), where $\frac d{dt}\frac{\partial T}{\partial\dot q_i}-\frac{\partial T}{\partial q_i}$ is traced to 1764, 24 years before the Mécanique Analytique. (Your answer would fit there just as well.) $\endgroup$ – Francois Ziegler Apr 13 '18 at 3:31
  • $\begingroup$ @FrancoisZiegler Yes, Euler can claim priority over Lagrange in its derivation. (In fact, Santilli 1978 p. 10 cites Euler 1736.) $\endgroup$ – Geremia Apr 13 '18 at 15:42
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Lagrange derived it in his Méchanique Analytique. Dugas summarizes it in his History of Mechanics pp. 342-344: Dugas p. 342 Dugas p. 343 Dugas p. 344

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