Was Hooke's law first coming from experiment or from math derivation of which Newtonian mechanical laws are the only prerequisite? Also can the law itself be reinvented in this way, or is it impossible to derive it mathematically from Newton's fundamental laws?
I want to know if Hooke's law is an axiom or a theorem, even though it was called a law...
Newtonian mechanics was not yet in place when Hooke published his De Potentia Restitutiva (On Restoring Force) in 1678, Newton's Principia only came out in 1687. Hooke inferred the law from experiments not only with springs but also with wood and a "body of air", concluding:
"The power of any spring is in the same proportion with the tension thereof: that is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward. Now as the theory is very short, so the way of trying it is very easie… it is very evident that… in every springing body… the force or power thereof to restore itself to its natural position is always proportionate to the distance or space it is removed therefrom".
This being said, Hooke did attempt a theoretical justification by imagining a mechanical model of springs as composed of tiny particles vibrating around some fixed positions, colliding with their neighbors and surrounded by "subtle matter". When the body is compressed the rate of collisions increases, accounting for the restoring force. When the body is stretched the force is provided by the pressure from "subtle matter". Although Hooke claims that the model leads to his law, modern analysis of his reasoning shows that it produces inverse square rather than proportional relationship. Today Hooke's law can be derived from more fundamental principles, but that requires assumptions about the molecular composition of matter and the nature of intermolecular forces, in addition to Newton's fundamental laws.
See Moyer's Robert Hooke's Ambiguous Presentation of "Hooke's Law".
There is no derivation of Hooke's Law from Newtonian mechanics. By Hooke or anyone else. At the time of Hooke, this was an experimental law. If Hooke had some theoretical explanation, it was incorrect. The derivation of this law from other more fundamental laws is a matter of Solid State physics, and I do not think this is achieved even today.
Remark: If "Newtonian mechanics" means the contents of his book Principia, then Hooke's law was published earlier.
The derivation from solid state physics is quite simple. At equilibrium, atoms are at distances where the first derivative of potential vanishes. So for small deviations, a series expansion is dominated by the quadratic term.
Force is the derivative of potential. Far a harmonic potential, the force increases linearly with distance from the equilibrium position. That is Hooke's law.
Hooke's law depends on the harmonic approximation of interatomic potential. He did not really know about atoms, but it seems that he was on to something. Moyer says that Hooke was wrong, but maybe Hooke was just mostly confused or lacking the terminology to express his intuition. It was before Brook Taylor introduced series expansions in 1715.
Once you have calculus and if we can assume that the tension is a smooth function of displacement, it is easy enough to discover Hookes law, it's simply the statement that to a good approximation that the tension is proportional to the displacement for small displacements. The physics comes in determining the constant of proportionality.
