How did Lagrange get his equation (the so-called Euler-Lagrange equation)

Question

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

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

Possible duplicate of who was the first to discover the Hamilton principle of classical physics? — Francois Ziegler, Apr 13 '18 at 03:05
@FrancoisZiegler This question appears to be a "how" question, not a "who" question. — Geremia, Apr 13 '18 at 03:11
@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.) — Francois Ziegler, Apr 13 '18 at 03:31
@FrancoisZiegler Yes, Euler can claim priority over Lagrange in its derivation. (In fact, Santilli 1978 p. 10 cites Euler 1736.) — Geremia, Apr 13 '18 at 15:42

score 4 · Accepted Answer · answered Apr 13 '18 at 03:10

4

Lagrange derived it in his Méchanique Analytique. Dugas summarizes it in his History of Mechanics pp. 342-344:

answered Apr 13 '18 at 03:10

Geremia

1 Answers1