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

Question

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 ✔️

Answer 1

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).

Answer 2

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).

Answer 3

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.

Answer 4

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.).

Answer 5

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.

Answer 6

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.

Answer 7

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.