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I understand both Newtonian and Lagrangian Mechanics are built upon some Principles. Besides Galileo's Principle of Relativity, Newtonian Mechanics counts with Newton's Laws of Motion and Lagrangian Mechanics counts with Hamilton's Principle (AKA the Principle of Least Action).

I am aware you can obtain Hamilton's Principle from Newton's Second Law and, using Lagrangian Mechanics, it is quite easy to prove Newton's First and Third Laws (and I guess it is equally simple with Newtonian Mechanics).

Therefore, my question is: how can one prove Newton's Second Law or what is the motivation for introducing it as a postulate?

I am assuming that the most fundamental block to build Mechanics is Newton's Second Law. If there is another way of building it eliminating any unproven or unmotivated statements, it would also answer my question. I did not consider building it from Hamilton's Principle because I believe there is no motivation to introduce it as a postulate instead of deriving it from Newton's Laws. If there is a motivation, this motivation would also answer my question.

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Newton's second law is not a good law unless you already have a definition of "mass". Otherwise saying $\vec{F}=\dot{\vec{p}}$ is just a definition. You should take the 3rd law into account, because that's what "fixes" it a little bit: you can define a standard mass, an so $m=m_s \frac{a_s}{a}$, and it only works with constant masses in intertial frames. So Newton laws work, but they're not too well formulated, you need something else.

Behind them, there's the principle of "conservation of momentum". Now, if we define what forces are (2nd law), then the 3rd law follows inmediately if momentum is conserved, so the key is conservation of momentum , which is fortunately explained by the Noether's theorems.

On the other hand, you can start saying that "Energy si constant" to get the same results.

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