What was the intuition behind the development of Hamiltonian mechanics?

[omie]

Is Hamiltonian mechanics a mathematical generalization of Newtonian mechanics or is it explaining some fundamental relationship that has a meaning that extends into our nature ? I guess my question is what would led William Rowan Hamilton to come up with his type of mechanics or anything relating to that.

 

[Omega0]

Is Hamiltonian mechanics a mathematical generalization of Newtonian mechanics

Yes

or is it explaining some fundamental relationship that has a meaning that extends into our nature ?

What is "extending into our nature" in the framework of physics? Isn't all physics describing nature?

I guess my question is what would led William Rowan Hamilton to come up with his type of mechanics or anything relating to that.

I absolutely can't speak for him. Historians may do a better job. What is about Langrangian mechanics? Do you think it is sort of step between?