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

Question

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.)

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.

Answer 1

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.

Answer 2

Kleppner's statement is carefully worded so that he is not claiming (1) that the third law follows from the first and second, nor (2) that the third law is necessary in order to make sense of the first and second laws. He's simply saying that there's a connection, not that they are conjoined twins. In my opinion he is still overstating the connection. For examples of experiments that directly test the third law, see Bartlett 1986 and Kreuzer 1968. Battat 2007 is a test of the first law that is independent of the third.

Kleppner says:

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.

The three experiments I've described all happen to be gravitational experiments. In principle, we could worry that a non-null result from an experiment such as Battat's could be interfered with by gravitational forces from distant bodies -- gravity is, after all, a long-range force. But these distant bodies would have to be unknown, or else we could account for their effects. If they were unknown, then Kleppner's proposed test doesn't work: we can't check whether they're accelerating due to the third-law partner of the force they exerted on our experiment.

Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21, summarized in Will, http://relativity.livingreviews.org/Articles/lrr-2006-3/

Battat 2007, http://arxiv.org/abs/0710.0702

Kreuzer, "Experimental measurement of the equivalence of active and passive gravitational mass," Phys. Rev. 169 (1968) 1007, http://bit.ly/13Z6XAm

Answer 3

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.

Answer 4

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.