forceshistorynewtonian-mechanics
For every action,there is equal and opposite reaction . This is the famous Newton's third law of motion. But how did he come to this conclusion? We can prove 2nd law using calculus. But how did Newton prove this law? Or did he just used practical examples?
[According to me, it may be due to conservation of energy,but did Newton use this?]
Newton's 1st and 2nd laws weren't particularly revolutionary or surprising to anyone in the know back then. Hooke had already deduced inverse-square gravitation from Kepler's third law, so he understood the second law. He just could not prove that the bound motion in response to an inverse square attraction is an ellipse.
The source of Newton's second law was Galileo's experiments and thought experiments, especially the principle of Galilean relativity. If you believe that the world is invariant under uniform motion, as Galileo states clearly, then the velocity cannot be a physical response because it isn't invariant, only the acceleration is. Galileo established that gravity produces acceleration, and its no leap from that to the second law.
Newton's third law on the other hand was revolutionary, because it implied conservation of momentum and conservation of angular momentum, and these general principles allow Newton to solve problems. The real juicy parts of the Principia are the specific problems he solves, including the bulge of the Earth due to its rotation, which takes some thinking even now, three centuries later.
The real history of scientific developments is complex, with many people making different contributions of various magnitude. The tendency in pedagogy is to relentlessly simplify, and to credit the results to one or two people, who are sort of a handle on the era. For the early modern era, the go-to folks are Galileo and Newton. But Hooke, Kepler, Huygens, Leibniz and a host of lesser known others made crucial contributions along the way.
This is especially pernicious when you have a figure of such singular genius as Newton. Newton's actual discoveries and contributions are usually too advanced to present to beginning undergraduates, but his stature is immense, so that he is given credit for earlier more trivial results that were folklore at the time.
To repeat the answer here: Newton did not discover the second law of motion. It was well known at the time, it was used by all his contemporaries without comment and without question. The proper credit for the second law belongs almost certainly to the Italians, to Galileo and his contemporaries.
But Newton applied the second law with genius to solve the problem of inverse square motion, to find the tidal friction and precession of the equinoxes, to give the wobbly orbit of the moon (in an approximation), to find the oblateness of the Earth, and the altitude variation of the acceleration of gravity g, to give a nearly quantitative model of the propagation of sound waves, to find the isochronous property of the cycloid, and a host of other contributions which are so brilliant ad so complete in their scope, that he is justly credited as founding the modern science of physics.
But in physics classes, you aren't studying history, and the applications listed above are too advanced for a first course, and Newton did indeed state the second law, so why not just give him credit for inventing it?
Similarly, in mathematics, Newton and Leibniz are given credit for the fundamental theorem of calculus. The proper credit for the fundamental theorem of calculus is to Isaac Barrow, Newton's advisor. Leibniz does not deserve credit at all. The real meat of the calculus however is not the fundamental theorem, but the organizing principles of Taylor expansions and infinitesimal orders, with successive approximations, and differential identities applied in varied settings, like arclength problems. In this, Newton founded the field.
Leibniz gave a second set of organizing principles, based on the infinitesimal calculus of Cavalieri. Cavalieri was Galileo's contemporary in Itali, and he either revived or rediscovered the ideas originally due to Archimedes in "The Method of Mechanical Theorems" (although he might not have had access to this work, which was only definitively rediscovered in the early 20th century. One of the theorems in Archimedes reappear in Kepler's work, suggesting that perhaps the Method was available to these people in an obscure copy in some library, and only became lost at a later date. This is pure speculation on my part. Kepler might have formulated and solved the problem independently of Archimedes. It is hard to tell. The problem is the volume of a cylinder cut off by a prism, related to the problem of two cylinders intersecting at right angles). Cavalieri and Kepler hardly surpassed Archimedes, while Newton went far beyond. Leibniz gave the theory its modern form, and all the formalism of integrals, differentials, product rule, chain rule, and so on are all due to Leibniz and his infinitesimals. Leibniz was also one of the discoverers of the conservation of mechanical energy, although Huygens has his paws on it too, and I don't know the dates.
The mathematicians' early modern history is no better. Again, Newton and Leibniz are given credit for theorems they did not produce, and which were common knowledge.
This type of falsified history sometimes happens today, although the internet makes honest accounting easier. Generally, Witten gets credit for everything, whether he deserves it or not. The social phenomenon was codified by Mermin, who called it "The Matthew principle", from the biblical quote "To those that have, much will be given, and to those that have not, even the little they have will be taken away." The urge to simplify relentlessly reassigns credit to well known figures, taking credit away from lesser known figures.
The way to fight this is to simply cite correctly. This is important, because the mechanism of progress is not apparent from seeing the soup, you have to see how the soup was cooked. Future generations deserve to get the recipe, so that we won't be the only ones who can make soup.
Newton's third law states that if object A acts on object B with force $\mathbf{F}_{AB}$, then object B must act on object A with force:
$$\mathbf{F}_{BA}=-\mathbf{F}_{AB}$$
When expressed in terms of A and B's momentum, the same equation can be written as:
$$\frac{\mathrm{d} \mathbf{p}_A}{\mathrm{d}t} = -\frac{\mathrm{d} \mathbf{p}_B}{\mathrm{d} t}$$
Rearranging:
$$\frac{\mathrm{d}}{\mathrm{d} t} \left [ \mathbf{p}_A + \mathbf{p}_B \right ] = 0$$
Or:
$$\mathbf{p}_A + \mathbf{p}_B = constant$$
This says that A and B must act on each other such that the sum of their momenta is constant over time. So Newton's third law is just an expression of the conservation of momentum. Momentum conservation itself can be seen as a consequence of spacial translation symmetry, as shown by Noether's Theorem. This symmetry remains in quantum mechanics, so momentum conservation still applies in QM.
Best Answer
This page has a helpful summary of the history--it seems he initially accepted the Aristotelian idea that objects could only continue to move if some "force" inside them was moving them (keep in mind this is before his technical definition of 'force'), and it took him a while to switch to the idea that bodies naturally tend to keep moving unless acted on by a force, i.e. inertia (it seems he got this idea from Galileo and Descartes). After this he developed the concept that "force" must be acting whenever there is a change in motion, i.e. an acceleration. From there, the article suggests he got the idea for the third law from various mechanical experiments in which it could be observed that the total momentum always remained constant (and if you define force as mass*acceleration, conservation of momentum implies that forces must always be equal and opposite):