# Proof of Newton's second law [duplicate]

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?

• What mathematical axioms would you start with to prove it? What mechanism would force nature to agree with your other axioms and your proof? – CuriousOne Apr 29 '15 at 3:20
• I'm not sure. That's why I asked the question! :) – Tdonut Apr 29 '15 at 3:21
• 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. – CuriousOne Apr 29 '15 at 3:27
• Possibly related: arxiv.org/abs/gr-qc/0612159 – Stan Shunpike Apr 29 '15 at 4:31
• No,2nd law cannot be proved. – Paul Apr 29 '15 at 5:05

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.