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.