# Are Newton's 1st and 3rd laws just consequences of the 2nd?

Can Newton's 1st and 3rd laws be assumed given just $F=ma$. I know that the argument would be, "No, then there would only be 1 law". But I can't think of any situation where 1 and 3 aren't superfluous.

If you just told me $F=ma$: I would assume nothing else causes an acceleration besides a force. So things not experiencing a force don't change velocity, even when velocity is 0. 1 ✔️

And, when two things that exist interact they use only their mass and acceleration to do so so they both must change in opposite ways. 3 ✔️

## 7 Answers

The position you are taking seems to depend on hindsight. Put yourself in the position of Newton being the first person to state these laws.

The first law was a flat-out statement that Aristotle was wrong when he stated that "nothing moves at all, unless a force which causes it to move is acting on it." Of course everybody now "knows" that Aristotle was wrong about that, so the "shock and awe factor" of Newton building his entire argument from that starting point no longer exists.

The second law then gives a definition of how to numerically measure the notion called "force." Of course it is consistent with the first law, since common sense would say that "no force" must have the measured value of $0$.

In modern terminology, the third law is a statement of the principle of conservation of momentum. It is independent of the first two laws - and apparently, the many crackpots who are still trying to invent perpetual motion machines and "free energy" devices still don't believe it is true, despite the empirical evidence (not to mention Noether's theorem).

• The first law is a definition of the intertial frames, no claim on change of motion is given - see my answer above. Aug 4 '16 at 9:03
• Poor Aristotle - one of the most brilliant minds to ever walk this planet and simply happened to arrive too early to be correct and thus we remember him as "That guy who tried to do physics and was wrong." Aug 4 '16 at 15:02
• @corsiKa I think it's fair to say that's the case for all physics. Someone tried to do it before, got stuff wrong, and then somebody else saw the problems and figured out a way to work them out. This is science at its best: a continual refinement of our understanding of the normal behavior of our universe. If Aristotle hadn't tried the wrong ways first, someone else would have, only later. Aug 4 '16 at 15:31
• @PyRulez Definitely. Einstein is also wrong, and so are the scientists who developed quantum theory. Each of those theories has problems as well. But this only strengthens my point: we are ever refining our understanding of the universe. It's just a question of, "How wrong?" and, "Under what conditions are these models still useful?" And in hundreds of years, people might think of Newton and Einstein in much the same way we do Aristotle. Aug 5 '16 at 22:20
• The other point to be made here is that while it is true that all physics is "wrong" in some sense, there are degrees of wrong, and in fact Aristotle is not 100% wrong. His physics is still good for basic movements of objects in the everyday environment on planet Earth. Objects sitting still on the Earth do not usually move, with respect to the Earth, unless you kick them. It's a reasonable model for simple everyday phenomena, it's just that it's not useful for much more than that. That is, its range of validity is very, very limited. But it's not zero. Aug 6 '16 at 3:46

Newton's first law defines the inertial reference frames:

• There exist in the universe some very particular reference frames such that, in those very particular reference frames (and only in those and no more) a body not subject to external forces or interactions moves with constant velocity.

Newton's second law states the change in motion in the above defined reference frames (and only in the above)

• In the above defined reference frames (see law 1) a body subject to external forces behaves as $$\textbf{F} = \dot{\textbf{p}}$$

I don't see how law 1 is a particular case of law 2, as law 2 is only valid after law 1 defines the inertial reference frames.

If you just told me F=ma...

the above is only valid in the reference frames defined by the first law.

And, when two things that exist interact they use only their mass and acceleration to do so so they both must change in opposite ways.

One does not know in principle how two things interact with each other. In particular that they must both change in opposite ways is a non-trivial statement. There is no a priori reason why it should be so (it could be anything else).

• I fail to see how the first law defines the inertial reference frames. Perhaps you could quote it and show how it is a definition of such? Thank you. Aug 4 '16 at 10:54
• @GennaroTedesco Do you have a citation for that? The usual statement of the first law is "An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force." ref Aug 4 '16 at 12:50
• I've never seen the first law stated like that. It is usually some variant on "An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force" or "When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force". I'm unable to read the original Latin, though. Aug 4 '16 at 12:51
• Newton himself could not possibly have stated his first law in terms of inertial frames, because the concept of inertial frames came later. You are perhaps correct to say that the first law can be understood to define inertial frames, but insisting that that is the only correct reading is both ahistorical and, IMNSHO, a poor pedagogical approach.
– zwol
Aug 4 '16 at 15:17
• @zwol Insisting to state Newton's laws as 5 hundreds years ago is in my opinion un-scientific. The content of the first law is to define the framework for the second law, whatever words one wants to use to describe them, that is the understanding of the law of motion. Aug 5 '16 at 7:10

No, the three laws are independent.

• The 1st law does not follow from the 2nd

The point here is that you have to understand what Newton meant by "force". For Newton, fictitious forces, i.e. forces which arise in accelerating reference frames, are not forces.

For example, if you are in an accelerating car, you will experience an acceleration in the direction opposite to that of the car's. But this acceleration is not caused by a (Newtonian) force.

So, for Newton, force implies acceleration, but acceleration does not imply force.

In modern terms, the first two laws would be formulated in the following way:

First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

Second law: In an inertial reference frame, the vector sum of the forces F on an object is equal to the time derivative of its momentum: $\vec F= \dot{\vec{p}}$.

Notice the first words in the statement of the second law: in an inertial reference frame. But what is an inertial reference frame? It is that which is defined by the first law. So, in modern terms, we would say that the first law defines the inertial reference frames, while the second law tells us how motion (momentum) and force are related in such frames.

Could we include fictitious forces in the second law to get rid of the first? Maybe. But this is not how Newton formulated the laws, and could result in a lot of complications.

For a nice discussion, see this article.

• The 3rd law does not follow from the second, either

If we consider a system of two point masses on which no external force is acting, we have, from the 2nd law (let's drop the vector notation for simplicity):

$$F=\frac{d}{dt} (p_1+p_2) =0 \to \frac{dp_1}{dt}+\frac{dp_2}{dt} = 0\\ \to F_1 = - F_2$$

Let's indicate with the notation $F_{ij}$ the force caused on particle $i$ by particle $j$. Since there is no external force, the force acting on particle 1 can come only from particle 2: $F_1=F_{12}$. The same is true for particle 2, so that we obtain

$$F_{12}=-F_{21}$$

So we were able to derive the 3rd law from the 2nd!

...Weren't we?

No. Consider now three particles: we would get

$$F_1+F_2+F_3 =0 \to (F_{12}+F_{13})+(F_{21}+F_{23})+(F_{31}+F_{32})=0$$

That is to say

$$\sum_{ij} F_{ij}=0$$

Of course, $F_{ij}=-F_{ji}$ (Newton's third law) is a solution of this equation...but it is not the only one!

So, Newton's third law is not a consequence of the second.

• Yo, that was mad helpful fool. So then did Newton just guess the third law or did he have some reason to believe this was the case? Aug 4 '16 at 13:52
• @BoddTaxter I think he just chose the easiest solution (Occam's razor!). Every other guess would be a bit arbitrary to say the least... PS: You can find a nice related discussion here: physics.stackexchange.com/questions/16162/… Aug 4 '16 at 14:05
• You prove that the second law implies the third by considering pairs and that considering triplets makes the third law consistent with the second law but you conclude that the third does not derive from the second? What kind of logics is that? Sep 8 '19 at 7:53
• @valerio How is it possible for an axiom to serve as definition? In general axiomatic systems have primitive notions (undefined terms) and axioms. Jun 15 at 19:15

The first law is a special case of the second law. The third does not follow from the second law. The third law states conservation of momentum. It holds if the system is described by a potential and the potential depends only on the relative positions of the bodies. For example, in one dimension for two bodies, the potential $U$ must be a function only of $x-y$ where $x,y$ are the positions. So for example with a potential $U(x,y) = Cxy$ the forces are not opposite and equal.

But note that no experiments have found violations of conservation of energy or momentum, so if you have such a model, it's a sign that you are throwing away some degree of freedom, some third body (e.g., heat, gravitational pull on the Earth, et.c.).

• "The first law is a special case of the second law" wrong. The first law defines the intertial frames. Once you define the intertial frames then you state the second law only in those frames. I can't believe there's still such confusion. Aug 4 '16 at 8:58
• @GennaroTedesco I don't see why you'd consider it weird - it's just one of those assumptions you get from hindsight. "Of course intertial frames are inertial, duh!" Most people don't realize that before that, inertial frames were not a known and defined concept (e.g. Aristotelian mechanics have no such concept). Aug 4 '16 at 13:34
• @Luaan I don't consider it "weird". It is just not what the first law is. Aug 4 '16 at 13:49
• @GennaroTedesco Where does the first law even mention the term "inertial frame"? Aug 4 '16 at 22:54
• @AntoniosSarikas Yes, of course. The point I want to make with my remarks to this answer is that it is false that first law is a special case of the second law. The second law only holds in inertial systems, which in turn are defined by the first law as those special systems where a body not subject to external forces moves in a straight line. This is how the laws of dynamics are to be intended and I find puzzling that some people still insist on nitpicking on Newton's definitions of 500 years ago. Jun 16 at 7:49

Newton's first law, a body remains in a constant state of motion or rest unless acted upon by a force, says that the proper reference frame for observing physics is an inertial frame. If you were on an accelerated frame then objects outside that frame would appear to accelerate without any measurable force. It is of course the case that if a force does interact with a body $F~=~ma$ tells you how that happens.

Newton's third law of motion, the change in momentum on one body is equal in magnitude and opposite in direction to that of a second body when they interact. This interaction can be a contact collision, a field or a spring or other mechanism. This tells us that $F~=~ma$ acts in space isotropically. We might say it tells us that space is isotropic. When combined with the first law it also tells us that a body changes its state of motion anywhere in space, so space is homogeneous. The reason is that a body can be in a constant state of motion, translating its position in space, and the second and third laws operate anywhere a force is present on that body. In a Noetherian sense the third law of motion gives us conservation of momentum.

This might add something to the preceding answers' discussion concerning the redundancy of the first law.

The two first laws, which relate the change of momentum of a body with the force applied to it, can apparently be summarized in a single equation:

$$\textbf{F} = \dot{\textbf{p}}$$

The first law is then just the special case in which $$\textbf{F}=\textbf0$$, $$\textbf{p}=const$$. This arguable tautology was pointed out by the physicist and philosopher Ersnt Mach in his book The Science of Mechanics, A Critical and Historical Exposition of its Principles:

Definition IV defines force as the cause of the acceleration, or tendency to acceleration, of a body. [...] We readily perceive that Laws I and II are contained in the definitions of force that precede. According to the latter, without force there is no acceleration, consequently only rest or uniform motion in a straight line. Furthermore, it is wholly unnecessary tautology, after having established acceleration as the measure of force, to say again that change of motion is proportional to the force. It would have been enough to say that the definitions premised were not arbitrary mathematical ones, but correspond to properties of bodies experimentally given.

The two first laws are, expressed succinctly:

1. A body will remain at rest or move with a constant velocity1 unless acted upon by a force.

2. Force is equal to mass multiplied by acceleration.

Mach's argument appears to be: one could express both of them in the single statement "If there is no force then there is no change in velocity" (which is just a consequence of the second law and the definition of acceleration). Then, a change in velocity can only occur if a force is present, but this is just what the first law tell us... then, it would seem that the first law is redundant. This view is also apparently shared by Harald Iro in his book A Modern Approach to Classical Mechanics.

However, I think there's a sense in which the first law can be considered an independent rule. I will quote a fragment of the discussion in this page, as I couldn't explain it better myself.

The key to unmasking the deception lies in understanding what Newton meant by "force". [...] For Newton, force is intimately connected with the frame of reference (or co-ordinate system) in which acceleration is measured. This leads to an important asymmetry: a force will cause an acceleration but an acceleration might not necessarily be caused by a force. An object can appear to accelerate when, in reality, it is the reference frame which is accelerating. For example, if I am seated in a train compartment then this is my reference frame. When the train leaves the station, then from my reference system, it is the train station that is accelerating away although, of course, no force is acting on the station.

If the reference frame is accelerating then a body otherwise at rest will appear to be accelerating away. Only in a framework which is stationary or moving with a constant velocity will a body remain at rest or move with a constant velocity unless acted upon by a Newtonian force. In all other frameworks a body will accelerate even when no force is present.

But the above phrase "a body [will] remain at rest or move with a constant velocity unless acted upon by a Newtonian force" is exactly Newton's first law. Therefore the first law defines the frames of reference in which Newton's concept of force is valid. They are frames of reference in which a body remains at rest or moves with a constant velocity unless acted upon by a Newtonian force. Such reference frames are called inertial frames of reference. All of Newton's three laws involve his concept of force so all three laws are only properly defined within inertial frames of reference.

I strongly recommend reading the rest of the article.

There's a modern re-interpretation of the first law in terms of differential geometry.

Here, velocities live in the tangent space and accelerations in the double tangent space. However, not arbitrary vectors of the double tangent space are valid accelerations - they neeed to be 'second order': If you think of the double tangent space as coordinated by $x, v, \dot x, \dot v$, we need to impose the condition $\dot x = v$. This means the zero vector of that space ($\dot x, \dot v = 0$) is in general not a valid acceleration.

A covariant connection can be used to lift velocities to corresponding 'zero' accelerations, and the first law can be understood as stating its existence. In non-inertial systems, the connection is non-trivial and responsible for the occurrence of 'fictious' forces.