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When doing a little internet search today on generalized coordinates I stumbled on this document:

http://people.duke.edu/~hpgavin/cee541/LagrangesEqns.pdf

If you are willing, would you be so kind as to open it up and look at the top of (numbered) page 6?

OK, so the very existence of this table tells me that these men formulated different ways to structure classical mechanics

I can accept Newton and d'Alembert as two different approaches (and in this post, when I use the name of the person, I assume the equation itself)

But I have difficulty seeing why this author lists Lagrange and Hamilton SEPARATELY.

(forget Gauss as that is not really relevant to my question.)

It seems to me in order to progress in mechanics, one MUST use Lagrange and Euler TOGETHER.

In other words, Hamilton provided a, well, blanket or superset to cover Lagrange. It really was not different (setting aside the Hamiltonian here and just looking at the two formulations of Lagranges equation and Least Action). Are those two not really to be taken TOGETHER? Am I missing something? Are Lagrange and Hamilton as distinct from each other as Newton is from d'Alembert?