It is going to be a long answer:
Newton's laws are not definitions. first let us see the apparent crazyness of Newton's laws which make them 'look like' definitions going in a circle, then we will resolve it (at least try to resolve it)
Let's see the introduction part of "Classical mechanics, the course of lectures"(1983) by A.K Raychaudhuri:
"At this stage, i.e. after learning Newton's laws of motion long ago and mastering its application in various fields, a student, who has been told quite often of the 'overthrow' of Newton's mechanics in our age, may be under the impression that he has learnt enough of Newtonian mechanics. 'I am a wise man and Newtonian mechanics is all but dead,' he may say, so what is the use in studying the subject once afresh ?'
If you think so, we shall try to make you sadder and perhaps a little wiser. We shall make no attempt to look at the twentieth century developments but we shall discuss the three laws of Newton which form the basis of Newtonian mechanics and just show that they are at least not as easy as they look at first glance.
Let us state the first law. It states:'Every body' continues in its 'state of rest' or of 'uniform motion' in a 'straight line' unless it is compelled by 'external forces' to change that state.
I have deliberately emphasized some words, and shall now try to analyse these.
Our first question is-what do you mean by 'every body'? At first sight just anything in the universe. But let me give an example. Suppose I put here a time bomb. It is at rest and we conclude that it will continue to remain so. But, at... hours and... minutes it explodes and the splinters run this way and that. Obviously, the body (i.e. the bomb or whatever exists of it) can no longer be said to be at rest as it was some time ago and no 'external force' was applied at any stage.
Is this an exception, indicating a breakdown of Newton's first law? Rather we see that we must define what exactly we mean by the term 'body' so as to exclude cases as the above.
Perhaps you could seek an escape by defining a 'body' (for the purpose of the first law) as 'one without any internal structure.' But that would require a definition of internal structure and such a definition seems not at all easy. At the empirical level, all macroscopic bodies (and microscopic too as the molecules, atoms and nuclei) have an internal structure. Thus Newton's first law would seem inapplicable to them all. And let me add that when you consider an 'elementary particle' like, say, the electron (which you may imagine has no internal structure), Heisenberg's uncertainty principle would come into play and then a body can be said to be at rest only if we ignore completely our knowledge about its position. (In fact, even for the classical electron, Newton's laws are not strictly obeyed. There is a self force term called the radiation reaction and thus the electron may spontaneously shoot out with increasing velocity.)
You may think that the introduction of the concept of centre of mass solves the problem- the centre of mass motion will obey Newton's first law. But the concept of centre of mass motion is by no means that simple - it involves the third law of Newton as also the constancy of mass (i.e. independence of velocity). Both these have their limitations.
Next comes the phrase 'state of rest'. A state of rest is essentially subjective (or relative, if you like). For example, the quiet boy who is at rest in my class is not at rest relative to an observer at, say the sun, for all the while he is moving in an ellipse along with the earth itself, to say nothing of other complicated motions of the earth.
So the question arises-what should be the frame of reference for the use of the phrase 'state of rest'? Would any reference frame be equally good for our purpose ?
Obviously not. Consider the case of two frames S and S', where S' is moving with uniform acceleration, say, f, with respect to the frame S. Suppose there is a body P which is at rest in S. We conclude that no force is acting on it, because otherwise P would be changing its state of rest. But now, what does an observer in frame S' find?
He sees that the body P is moving with a uniform acceleration -f, or, in other words, it is neither at rest nor in a state of uniform motion in a straight line. But we have already seen that there is no force on P, so how can it be in an accelerated motion, if Newton's first law is to be valid? Thus, the law cannot be valid in both frames; if it is true in S, it is false in S'
So we conclude that Newton's first law of motion is not valid for all frames of reference-at most it is valid for some frames, which are incidentally called Inertial Frames. Newton's law thus simply asserts the existence of Inertial Frames.
The first question that arises now is-do such frames really exist? Or, in other words, is the first law true in any frame whatsoever? Operationally it does not seem possible to attain a condition where a body is subject to no force whatsoever. One has to be content to a degree of approximation. The second question is to find out observationally the preferred frames which we are calling inertial frames. This, as you probably know, is the objective of the Foucault pendulum and gyroscope experiments. The third question, which is rather philosophical, is why are some frames or states of motion specially favoured from the point of view of this particular law of physics ?
The second and third questions have led to the so-called Mach's principle that inertia is not an intrinsic property of bodies but arises out of interaction between different particles of matter in the entire universe. The principle is a difficult one and we cannot enter into its discussion. We can simply note that it plays an important role in some areas of modern physical thought.
Our next question is about uniform motion. We define uniform motion to be one in which a body traverses equal distances in the same direction in equal intervals of time. This looks simple enough, but we ask what precisely do we understand by equal intervals of time ?
It will not do to say that a nice chronometer will settle our problem. For how do you guarantee about its performance? Of course there are ways out. For example we may assume the velocity of light to be a constant in time and thus equal intervals of time are those in which light goes through equal distances. But this is just an empirical assumption. Alternatively as the frequency of a spectral line can be linked with time, we define time intervals with the help of spectral lines (as is indeed the practice now). But as our theory shows these frequencies depend on the Planck constant h and the masses and charges of the elementary particles. We are thus assuming that these do not undergo any secular variation.
We may even say that if no force acts on a body, the time intervals, in which it traverses equal distances, are equal. Looked at this way, this part of the first law is only a definition of equal intervals of time.
The next topic is the straight line. How can one be sure that a line is straight? If you say that it is just the line of shortest length between two points, the question is: how do you operationally verify it? Again you may say that the straight line is the path taken by a free particle. But then a part of the first law reduces to the definition of a straight line. Or you could declare that a straight line is the path that light takes but that only shows that Newton's law as it stands is incomplete and has to be supplemented by the law of light propagation.
Lastly we come to force. Initially the concept of force was a very intuitive one. It was a push or a pull. Let us consider a piece of chalk. If placed on the table, it remains at rest.
Now suppose I lift the chalk to some height and just release it. It will begin to fall and if we care to measure, the motion is accelerated. How is it possible, for apparently there is neither any push nor pull on the chalk ?
You will say that there is the force of gravity acting on the piece of chalk. Well, I ask-what is the proof of existence of such a force ? You argue that the acceleration of the piece of chalk coupled with Newton's first law shows the existence of such a force. Thus the whole argument goes on in a circle. Should we not rather say that we have taken Newton's laws as sacred, and just to save them, have affirmed the existence of a gravitational force? In a very similar way, we were led to discover electromagnetic forces to save Newton's laws.
Now let us suppose that a Sadhu has acquired such a power that he can sit quietly in air without any support. This is the phenomenon called levitation. Suppose too that the Sadhu is no fraud and he can really do the trick. Then what is the situation? A body is in a state of rest although a force (the force of gravity) acts on it.
We could say that there are supernatural phenomena where Newton's laws do not apply. But I think we would prefer a different approach. We would bring in a new force-the Yogic force (say)-into our physics. We would say that the Sadhu has been able to balance gravity with the help of this Yogic force. That saves Newton's laws again. We would then proceed to find the sources of this new force, determine the laws, assuming hopefully that such laws do indeed exist, and finally proceed to find out the field equations. Perhaps the enterprising theoretical physicist would attempt a quantization of the field!
If it brings derisive smiles to your lips, I can only tell you that I am recapitulating the history of physics in a hypothetical context.
But with that comes the problem-how can we be sure as to whether any force is acting on a body or not? As long as the concept of force was restricted to push and pull, the answer was simple. But with the introduction of other kinds of forces, the question is not at all simple. One can say, 'Just measure whether the motion of the body is accelerated or not; if you find an acceleration, you know that there is a force.'
But in that way Newton's law is only a definition of force and as such unverifiable. What we would have liked was to see whether there is any acceleration, and observe independently whether any force acts, and lastly combine the observations to test Newton's law.
So much for the first law. We pass on next to the second law which states, 'the force vector acting on a body is proportional to the rate of change of the momentum vector of the body'. In mathematical form (in an obvious notation),
F= dp/dt....(1)
where we have replaced the expression 'proportional to' by 'equal to' with a suitable choice of units.
We define the momentum vector as one proportional to the velocity vector.
p=mv.....(2)
Here m is a constant for the body in question (assumed independent of velocity in non-relativistic mechanics). This m is called the 'mass' or 'inertia' of the body. We may combine equations (1) and (2) to get :
F = d/dt (m.v) ....(3)
But this is confusing. For this last equation attempts to embody the definition and measurement of two quantities, viz., the mass and the force. One single equation cannot do that. So we get hardly anything from the second law.
At this point the third law comes to our rescue. 'To every action there is an equal and opposite reaction'. Let us consider that there are two particles 1 and 2. F(12) denotes the force on particle 1 due to particle 2; then the third law states :
-F(12)= F(21).......(4)
Combining this with the second law (cf. equation 3), we get (in a system where no force acts from outside):
d/dt (Σmv) =0
or, Σmv = constant; we say that the momentum is conserved.
For a system consisting of only two particles (velocities all in one direction)
m(1)v(1)+m(2)v(2)= m(1)v'(1)+m(2)v'(2)
Or, m(1)/m(2) = - [v(2)-v'(2)] /[v(1)-v'(2)]
The notation is obvious and we can see how by measuring the changes of velocities, we can compare the masses. Thus mass being independently determined, equation (3) suffices to specify the force.
However, difficulties do remain, for the third law is not always correct-a simple example is that of two current-carrying wires, if the currents be not parallel. (However if one considers two complete circuits, the total forces are equal and opposite.)
If I have been able to impress anything upon you, you must have appreciated that physics is essentially conservative and a law, once it is formulated, is not given up easily. Rather, the definitions are changed to fit in with new discoveries.
So when you find that the third law has limitations and consequently symbol Σmv may not remain constant, you do not throw away the conservation of momentum principle but rather begin to call some other quantities as momentum to save your principle. Thus we had the concept of momentum of electromagnetic fields or indeed of any field whatsoever. Still difficulties remain -the third law apparently implies action at a distance with infinite speed of propagation.
To conclude, in this lecture I have not attempted a complete analysis of the questions raised. That of course does not mean that all the questions are unanswerable . But some atleast, are unanswerable within the Newtonian frame work."
Now see what Richard Feynman has to say about this matter in lecture of physics (Vol:1) I hope that will resolve your issue with Newton's law.
"Although it is interesting and worthwhile to study the physical laws simply because they help us to understand and to use nature, one ought to stop every once in a while and think, “What do they really mean?” The meaning of any statement is a subject that has interested and troubled philosophers from time immemorial, and the meaning of physical laws is even more interesting, because it is generally believed that these laws represent some kind of real knowledge. The meaning of knowledge is a deep problem in philosophy, and it is always important to ask, “What does it mean?”
Let us ask, “What is the meaning of the physical laws of Newton, which we write as F=ma? What is the meaning of force, mass, and acceleration?” Well, we can intuitively sense the meaning of mass, and we can define acceleration if we know the meaning of position and time. We shall not discuss those meanings, but shall concentrate on the new concept of force. The answer is equally simple: “If a body is accelerating, then there is a force on it.” That is what Newton’s laws say, so the most precise and beautiful definition of force imaginable might simply be to say that force is the mass of an object times the acceleration. Suppose we have a law which says that the conservation of momentum is valid if the sum of all the external forces is zero; then the question arises, “What does it mean, that the sum of all the external forces is zero?” A pleasant way to define that statement would be: “When the total momentum is a constant, then the sum of the external forces is zero.” There must be something wrong with that, because it is just not saying anything new. If we have discovered a fundamental law, which asserts that the force is equal to the mass times the acceleration, and then define the force to be the mass times the acceleration, we have found out nothing. We could also define force to mean that a moving object with no force acting on it continues to move with constant velocity in a straight line. If we then observe an object not moving in a straight line with a constant velocity, we might say that there is a force on it. Now such things certainly cannot be the content of physics, because they are definitions going in a circle. The Newtonian statement above, however, seems to be a most precise definition of force, and one that appeals to the mathematician; nevertheless, it is completely useless, because no prediction whatsoever can be made from a definition. One might sit in an armchair all day long and define words at will, but to find out what happens when two balls push against each other, or when a weight is hung on a spring, is another matter altogether, because the way the bodies behave is something completely outside any choice of definitions.
For example, if we were to choose to say that an object left to itself keeps its position and does not move, then when we see something drifting, we could say that must be due to a “gorce”—a gorce is the rate of change of position. Now we have a wonderful new law, everything stands still except when a gorce is acting. You see, that would be analogous to the above definition of force, and it would contain no information. The real content of Newton’s laws is this: that the force is supposed to have some independent properties, in addition to the law F=ma; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law F=ma is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple.
Now the first example of such forces was the complete law of gravitation, which was given by Newton, and in stating the law he answered the question, “What is the force?” If there were nothing but gravitation, then the combination of this law and the force law (second law of motion) would be a complete theory, but there is much more than gravitation, and we want to use Newton’s laws in many different situations. Therefore in order to proceed we have to tell something about the properties of force.
For example, in dealing with force the tacit assumption is always made that the force is equal to zero unless some physical body is present, that if we find a force that is not equal to zero we also find something in the neighborhood that is a source of the force. This assumption is entirely different from the case of the “gorce” that we introduced above. One of the most important characteristics of force is that it has a material origin, and this is not just a definition.
Newton also gave one rule about the force: that the forces between interacting bodies are equal and opposite—action equals reaction; that rule, it turns out, is not exactly true. In fact, the law F=ma is not exactly true; if it were a definition we should have to say that it is always exactly true; but it is not.
The student may object, “I do not like this imprecision, I should like to have everything defined exactly; in fact, it says in some books that any science is an exact subject, in which everything is defined.” If you insist upon a precise definition of force, you will never get it! First, because Newton’s Second Law is not exact, and second, because in order to understand physical laws you must understand that they are all some kind of approximation.
Any simple idea is approximate; as an illustration, consider an object, … what is an object? Philosophers are always saying, “Well, just take a chair for example.” The moment they say that, you know that they do not know what they are talking about any more. What is a chair? Well, a chair is a certain thing over there … certain?, how certain? The atoms are evaporating from it from time to time—not many atoms, but a few—dirt falls on it and gets dissolved in the paint; so to define a chair precisely, to say exactly which atoms are chair, and which atoms are air, or which atoms are dirt, or which atoms are paint that belongs to the chair is impossible. So the mass of a chair can be defined only approximately. In the same way, to define the mass of a single object is impossible, because there are not any single, left-alone objects in the world—every object is a mixture of a lot of things, so we can deal with it only as a series of approximations and idealizations.
The trick is the idealizations. To an excellent approximation of perhaps one part in 1010, the number of atoms in the chair does not change in a minute, and if we are not too precise we may idealize the chair as a definite thing; in the same way we shall learn about the characteristics of force, in an ideal fashion, if we are not too precise. One may be dissatisfied with the approximate view of nature that physics tries to obtain (the attempt is always to increase the accuracy of the approximation), and may prefer a mathematical definition; but mathematical definitions can never work in the real world. A mathematical definition will be good for mathematics, in which all the logic can be followed out completely, but the physical world is complex, as we have indicated in a number of examples, such as those of the ocean waves and a glass of wine. When we try to isolate pieces of it, to talk about one mass, the wine and the glass, how can we know which is which, when one dissolves in the other? The forces on a single thing already involve approximation, and if we have a system of discourse about the real world, then that system, at least for the present day, must involve approximations of some kind.
This system is quite unlike the case of mathematics, in which everything can be defined, and then we do not know what we are talking about. In fact, the glory of mathematics is that we do not have to say what we are talking about. The glory is that the laws, the arguments, and the logic are independent of what “it” is. If we have any other set of objects that obey the same system of axioms as Euclid’s geometry, then if we make new definitions and follow them out with correct logic, all the consequences will be correct, and it makes no difference what the subject was. In nature, however, when we draw a line or establish a line by using a light beam and a theodolite, as we do in surveying, are we measuring a line in the sense of Euclid? No, we are making an approximation; the cross hair has some width, but a geometrical line has no width, and so, whether Euclidean geometry can be used for surveying or not is a physical question, not a mathematical question. However, from an experimental standpoint, not a mathematical standpoint, we need to know whether the laws of Euclid apply to the kind of geometry that we use in measuring land; so we make a hypothesis that it does, and it works pretty well; but it is not precise, because our surveying lines are not really geometrical lines. Whether or not those lines of Euclid, which are really abstract, apply to the lines of experience is a question for experience; it is not a question that can be answered by sheer reason.
In the same way, we cannot just call F=ma a definition, deduce everything purely mathematically, and make mechanics a mathematical theory, when mechanics is a description of nature. By establishing suitable postulates it is always possible to make a system of mathematics, just as Euclid did, but we cannot make a mathematics of the world, because sooner or later we have to find out whether the axioms are valid for the objects of nature. Thus we immediately get involved with these complicated and “dirty” objects of nature, but with approximations ever increasing in accuracy."
