What exactly did Lagrange do, historically? This is very relevant but the accepted answer just does similar speculations based on contextual details that I have done. It was mentioned that part of Lagrange's research programme was to remove all mathematical dependence on geometric interpretation. He thinks the title Mecanique Analytique is a reference to that. This is a very useful insight into the mind of Lagrange when developing his formulation but I'm looking for something more. preferably from pen or tongue of the inventors themselves.
Today we often study these formulations and their appropriate circumstances of usage and may become quite skilled at applying to them interesting problems but I think most rarely consider how the newtonian, and now the other formulations were developed.(i think there is a tendency to dismiss the roots of milestone achievements as the product of pure genius incomprehensible to mortal men like us)
More specifically, I mean the processes of ideation that led to the mathematical construction of these formulations. Googling around, I have read that it is unknown if Lagrange said about the developmental process of his mechanics, and the same goes for newton but even more so since he had a reputation for being quite prickly on the topic of sharing his mental processes that led to a discovery. So I have little hope that there is anything newton wrote or said relevant to my question, but perhaps he let something slip and that's why i'm asking here.
I could not find anything about hamilton, and I am also not yet desperate enough to read all the extant works of these mechanicians in the hopes of gleaning something.
If they all chose to remain silent, then the only thing we can do is:
Know the common objective of these theories. That is, determining the a(t) v(t) r(t) of a moving body based on the dynamical portions of its motion.
Study the development of mechanical theory until the individual in question.
In the case of Newton, models like kepler's laws existed for example. Whereas in the case of Lagrange, Newtonian mechanics already existed.
Compare the most appropriate circumstances of usage. This is particularly helpful with cases like Lagrange and Hamilton, because they must have saw a need to develop a new formulation despite the existing ones. Otherwise why bother? They must have been faced with problems where the previous formulations were too cumbersome.
So by comparing what lagrangian mechanics is more efficient at than newton's we could glean that the mind of lagrange once wrestled with the idea of:
"if only there was a less cumbersome way of solving this pendulum problem and similarly constrained problems..sigh. If only." invents lagrangian
Study what has been transmitted from their lives until they published their formulations.
But I hope it wont be necessary to speculate this hard, as hopefully the historians here are aware of some rare transmissions from these inventors about how they ideated their formulations.