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Is it possible to prove Newton's second law of motion, $F=ma$? Or is it just realized because of proportionality and the definition of a Newton of force?

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    $\begingroup$ What mathematical axioms would you start with to prove it? What mechanism would force nature to agree with your other axioms and your proof? $\endgroup$ – CuriousOne Apr 29 '15 at 3:20
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    $\begingroup$ I'm not sure. That's why I asked the question! :) $\endgroup$ – Tdonut Apr 29 '15 at 3:21
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    $\begingroup$ With regards to the first question, I think that Landau and Lifshitz have given a beautiful derivation of it in "Course on Theoretical Physics Volume I, Mechanics", where they start from a variational principle from which one can then "derive" Newton's laws. Does nature care? No. Classical mechanics is a 100% false description of nature. It can't explain matter and light and it gives the wrong explanation for gravity. Other than that, of course, it works almost flawlessly on the scale of human beings to roughly the scale of the solar system and maybe a bit beyond. $\endgroup$ – CuriousOne Apr 29 '15 at 3:27
  • $\begingroup$ Possibly related: arxiv.org/abs/gr-qc/0612159 $\endgroup$ – Stan Shunpike Apr 29 '15 at 4:31
  • $\begingroup$ No,2nd law cannot be proved. $\endgroup$ – Paul Apr 29 '15 at 5:05
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Newton's second law is a generalization of experience. It has no derivation in simpler terms.

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Nothing in physics can be proven in the mathematical sense. Sometimes mathematical physicists begin with axioms that are thought to model an aspect of the natural world and then work out what follows from these axioms. Their derivations are proofs in the mathematical sense of what follows from the axioms, but ultimately they do not prove anything about the natural world. If their proofs fail to reflect observed reality, however, we can say (contrapositively) that at least one of the beginning axioms must be untrue.

An example of such theorems (in this special sense of the word) is the spin statistics theorem.

Newton's law on the other hand is more of a definition made to describe our experience. Beginning with Newton's first law, that something's state of motion does not vary with time unless it interacts with something else, we seek to describe more precisely the case where there is such interaction and therefore a change in motion state. It is natural to say that the more the first law is violated, the stronger must be the "interaction". We also witness that the same causative agent of such a "violation" (e.g. a stretched spring) affects different bodies differently. Bodies that are "more massive" in some way are affected less than "less massive" ones.

So one can encode these qualitative, but accurate, descriptions in a definition $\vec{F}=\mathrm{d}_t (m\,\vec{v})$. So $\vec{F}$, by definition, encodes the "strength" of the interaction that begets a change $\mathrm{d}_t$ of state of motion $m\,\vec{v}$. The stronger the interaction, the swifter the deviation from the First-Law-foretold motion. The constant of proportionality $m$ quantifies how different things react to the same causative agents: if something's acceleration is twice as much as that of something else under the influence of the same causative agent (e.g. the same stretched spring), then we say by definition that the latter is twice as massive as the former.

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