In 1833 Hamilton introduced what today is called the Hamiltonian formalism in classical mechanics.
I am wondering what was his motivation. Did he try to solve a specific question?
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The page titled On a general method in dynamics is specifically about Hamilton's work on the subject of mechanics.
The articles by William Rowan Hamilton do not contain equations in the form we know them today.
Jacobi later stated that Hamilton gave two equations, and that one of those two was superfluous, so in his own work Jacobi omitted that one.
About Hamiltonian formalism:
I'm not sure, but I don't think that what today is referred to as 'Hamiltonian formalism' is present in Hamilton's articles.
Rather, I suspect that what is referred to as Hamiltonian formalism was introduced at a later date, and attributed to Hamilton.
I want to emphasize:
In history of science the only reliable sources are the original works, and articles and books by historians of science who use as source the original works exclusively.
Conversely: the 'historical introduction' sections in physics textbooks should not be trusted.
My impression is that textbook authors write such introductions from memory, reconstructing how they remember the 'historical introduction' in the textbook they grew up with. I very much suspect that each time the story is retold the author embellishes the story.
It's not that such embellishment done on purpose. For sure the authors are not aware of what they are doing. It's just: there are circumstances where human memory is very fallible.
Mark Twain wrote the following quip:
"In the real world, nothing happens at the right place at the right time. It is the job of journalists and historians to correct that."
Textbook authors tend to write the story as they remember taking it in, but some elements are lost, and other elements are added, and the story is reshaped to a form that aligns with the author's expectation.
So that is why I assert:
Only the original works are reliable as source of history of science.
Although I believe the answer by Cleonis contains much general wisdom on the subject of myths and fictional embellishments in science and math history, still on this occasion some relevant papers by Hamilton have been overlooked.
See Hamilton's "On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function" in Philosophical Transaction (1834) vol.124 p 247-308.
There is a further important paper, "Second essay on a general method in dynamics" in 1835.
(There are plenty of equations in there, that seem not too far away from the modern forms, but I can't give an analysis.)