# Can one of Newton's Laws of motion be derived from other Newton's Laws of motion?

Can one derive Newton's

second and third laws from the first law or

first and third laws from the second law or

first and second laws from the third law

I think Newton's laws of motions are independent to each other. They can not be derived from one another. Please share the idea.

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–  Qmechanic May 27 at 12:28

The modern interpretation of Newton's First Law is about the existence of inertial reference frames, mainly to solidify the idea that such coordinate systems exist and are important.

However, I sincerely doubt this is what Newton himself had in mind when he postulated his laws. Historically, Newton probably introduced his first law to put emphasis on the fact that moving bodies do not slow down by their own accord (as was common wisdom at the time). So yes, the first law in this context can certainly be derived from the second law by setting F=0. In fact I doubt Newton even had an idea of what an inertial frame was (though it probably could have been explained to him with relative ease).

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+1 for carefully considering the various inequivalent formulations of the first law –  Ben Crowell May 27 at 15:08

Newton's laws of motion cannot be derived from each other. They are the building blocks of Newtonian mechanics and if fewer were needed, Newton would simply formulate fewer.

The first law postulates the existence of an inertial reference frame in which an object moves at constant velocity if the net force acting on it is zero. Although it might seem you can derive it from the second law (if the net force is zero, there is no acceleration and the velocity is constant) but in fact, both second and third law assume that the first law is valid. If an observer is in a non-inertial reference frame, she will observe that the second and third laws are not valid (when you sit in an accelerating car, the Earth accelerates in the opposite direction without any force acting on it).

You also cannot derive the second law from the first one because all you know from the first law is that when an object accelerates, there is a force acting but the first law says nothing about the relation between the force and the acceleration. That's what second law is for, to say that there is a linear relationship.

The third law adds something more to the first and second laws. It deals with interactions and states that two bodies exert same but opposite forces o each other. That is something you cannot see from the first or second law and similarly, there is no way to use this to derive the second law (you cannot derive the first law because that is assumed to be valid in order to postulate the third law).

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"They are the building blocks of Newtonian mechanics and if fewer were needed, Newton would simply formulate fewer." The version of the first law that you give is not the one Newton wrote. It's fine to say that this modern reformulation of Newton's laws has no redundancy, but it doesn't make sense to impute that lack of redundancy to Newton. In Newton's original formulation, the first law is clearly a special case of the second. –  Ben Crowell May 27 at 14:57
@BenCrowell: is there a scientific consensus on the fact that law 1 would be a specific case of law 2 in Newton's mind? –  pluton Aug 27 at 2:42

No, they're not independent, because the first can be deduced from the second. Newton's second law says that $F=ma$. The first law says that $a=0$ when $F=0$, which clearly follows from $F=ma$. The purpose of the first law is not to be an independent postulate from the second law, but just to emphasise this particular special case, which presumably would have been counterintuitive to many of the contemporary readers of Newton's work.

Other answers have tried to claim that the first law is really about the existence of an inertial reference frame, and of course you're free to interpret it that way if you want, but what it actually says is not independent of the second law.

From a modern point of view, all of newton's laws follow from the conservation of momentum. For example, for two bodies in one dimension, the total momentum is $m_1v_1 + m_2v_2$. If it doesn't change over time then its derivative must be zero, i.e. $$\frac{d}{dt}(m_1v_1 + m_2v_2) = m_1a_1 + m_2a_2 = 0.$$ If we define $F_1 = m_1 a_1$ and $F_2 = m_2 a_2$ then this becomes $F_1 = -F_2$, which is Newton's third law. The second law is just the definition of $F$, and the first law comes from noting that if you just have one body then $mv$ can't change, so $v$ has to be constant.

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I disagree - from a geometric point of view, the important part of the first law is the existence of a notion of being at rest or of moving uniformly straight forward; I'm actually reading through Giachetta, G.; Mangiarotti, L.; Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics right now, where this translates to the choice of a holonomic connection on the jet bundle $J^1Q\to\mathbb R$, whereas external forces are given by sections of the vertical cotangent bundle of $J^1Q\to Q$ –  Christoph May 27 at 11:25
@Christoph I can't be sure, because I haven't read the original, but in the context of Newton's work, I think the important point was much more likely "it won't stop moving if you stop pushing it." Although Newton didn't discover this (Galileo certainly knew it), it was still a relatively new idea. I doubt Newton would have known a holonomic connection on a jet bundle if you'd hit him over the head with it. As I said, you can interpret it that way if you want, but in terms of what the first law actually says, it's just a special case of the second. –  Nathaniel May 27 at 12:23
It's fine to consider the second law as a definition rather than a physical law, but it still ends up having predictive value because we have other ideas about forces that we've already established, e.g., that force is what we measure on a spring scale and that the same force can be applied by the same means to different objects. I think you also need conservation of mass, because otherwise there's no reason to believe that mass is a fixed property of a material object. After all, you can's use $F=ma$ to define both $F$ and $m$. –  Ben Crowell May 27 at 15:14