# Logical connection of Newton's Third Law to the first two

The first law and second laws of motion are obviously connected. But it seems to me that the third law is not related to the first two, at least logically.

(In Kleppner's Mechanics the author states that the third law is a necessity to make sense of the second law. It didn't make sense to me, though. I'll post the excerpt if anyone would like to see it.)

Thanks, Ron

EDIT: Excerpt from Introduction to Mechanics by Kleppner & Kolenkow (1973), p. 60:

Suppose that an isolated body starts to accelerate in defiance of Newton's second law. What prevents us from explaining away the difficulty by attributing the acceleration to carelessness in isolating the system? If this option is open to us, Newton's second law becomes meaningless. We need an independent way of telling whether or not there is a physical interaction on a system. Newton's third law provides such a test. If the acceleration of a body is the result of an outside force, then somewhere in the universe there must be an equal and opposite force acting on another body. If we find such a force, the dilemma is resolved; the body was not completely isolated. [...]

Thus Newton's third law is not only a vitally important dynamical tool, but it is also an important logical element in making sense of the first two laws.

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Hi Ron, and welcome to Physics Stack Exchange! I think it would help if you can quote the piece from Kleppner that you're talking about, as long as it's not too long. – David Zaslavsky♦ Dec 11 '11 at 3:21
Thanks! I had already edited the original post and added the excerpt I hope it isn't too long. – Ron Dec 11 '11 at 6:06
It is not clear to me, what's exactly your question. – student Dec 11 '11 at 9:52
There is a historical context which hasn't been address by other answers to date, and may play an important part in the seemingly redundant statement of the first and second laws. Previous to Newton, much though on the nature of motion in Europe could be traced back to Aristotle, who claimed that bodies in motion naturally tended to return to rest (nor is that stupid: try sliding a book on a table). The first law is a bald contradiction of this doctrine. – dmckee♦ Dec 11 '11 at 17:45

## 3 Answers

Newton's Fist and Second Law relate forces acting on a single system to conservation or changes of that system's momentum. It doesn't say anything about the nature of these forces, their origin; they could come "out of nowhere" and Laws 1 and 2 still hold.

The Third Law, however, indicates that all forces, or "actions", are just one side of an interaction. This view, that systems act on each other, by either attracting or repelling each other, still holds even in circumstances where other aspects of newtonian mechanics don't (i.e. relativistic or quantum mechanics). All forces observed since Newton's time until today are still modelized as being the results of the fundamental interactions.

One could say that the First Law describes the nature of momentum, the Thid Law the nature of forces, and the Second Law the link between the two. The First and Third Law thus provide the setting where the Second Law is stated.

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Wow! Thanks! This post made a lot of sense. I agree. Newton, as you have posted, describes the nature of momentum and also, I think, in hindsight he gives a definition of a force, i.e. that which changes the  momentum of the object. – Ron Dec 12 '11 at 11:08

Newton's third law of motion gives meaning to the first two laws by restricting what type of fundamental forces act between particles. This restriction gives meaning to the "force" described in the first two laws.

Taken by themselves (by which I mean without any reference to any explicit form for the fundamental forces acting between particles), the third law is what gives the first two laws any predictive power. This is what Mr. Kleppner is referring to when he says that Newton's third law is "an important logical element in making sense of the first two laws."

For example, suppose you are watching two balls float in outer space; ball A and ball B. You see ball A accelerate towards ball B. Using Newton's first law you know there is a force acting on ball A from ball B. Using Newton's second law you know that the force is along a vector connecting the two balls. Finally, using Newton's third law you can predict that ball B should also be accelerating towards ball A. And you can test that prediction. So, the third law is what gives the first two laws any predictive power.

I would argue that instead of Newton's third law it is the explicit form of the fundamental forces which "really" give the first two laws their meaning. In this case, Newton's third law of motion is just a restriction on what form those fundamental forces can take.

To illustrate this idea, suppose we want to use Newton's laws to do some science.

Newton's first law states:

A body in motion will stay in motion unless acted upon by an external force.

Since we don't know what a force is yet, it could be anything and this statement can be rephrased as:

A body in motion will stay in motion unless it doesn't.

You can see why this is not useful.

Newton's second law of motion states:

The acceleration of a body is parallel and proportional to the force exerted on the body and inversely proportional to the mass of the body.

Again, without a definition of force, this statement is useless.

Now, suppose we have a definition for a force, such as gravity.

$F_{gravity} = G \frac{m_{1} m_{2}}{r^2}$

Now, the first two laws have meaning. We can use our definition of force to predict the motion of a particle due to the gravitational attraction of some other body and then go out and test it!

Newton's third law of motion states:

Any force exerted by body A on body B implies an equal and opposite force on body B by body A.

Given a description of all the fundamental forces acting between particles, we don't need Newton's third law. Instead, Newton's third law is telling us how these forces act, namely symmetric with respect to both particles. You can see this reflected in the mathematical formula for the force of gravity; switching $m_1$ and $m_2$ you get the same force.

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I think you make here two interesting points: Newton's second law is an incomplete law (it contains a definitional element, you still have to work out the explicit form the force takes in a particular circumstance) and Newton's third law poses some restrictions on the mathematical form of the force, since interactions between two particles are symmetric with respect to both. My question is: does Newton's third law incorporate somehow the classical principle of relativity? – quark1245 Dec 11 '11 at 9:32

Today, Newton's first law is often interpreted (or extended) to the statement that inertial reference systems exist. For example let me cite from Jose,Saletan:"Classical Dynamics A contemporary Approach"

There exist certain frames, called inertial, with the following two properties.

Property A) Every isolated particle moves in a straight line in such a frame
Property B) If the notion of time is quantified by defining the unit of time so that one >particular isolated particle moves at constant velocity in this frame, then every other >isolated particle moves at constant velocity in this frame. "

Otherwise the first law would be a trivial consequence of the second one...

Concerning the connection between the second and the third law, note that one has to define the word "mass". This is sometimes done via the third law. If you do so, you need the third law just to understand the variables that occur in the second law.

Note that there where many many critiques in the history of physics concerning the locigal status of Newton's laws and there were many attempts to make it logically clearer.

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