THE CLASSICAL MECHANICSFebruary 1, 2022
PART 3: FORCE
The English teach mechanics as an experimental science. Here on the Continent, it is taught always as a deductive and à priori science.
The English are right.
Why did the Continental scientists stay so long in theory?
On the other hand, if the principles of mechanics are only experimental, then they are merely approximate and provisory. It means we can have new experiments to modify or abandon them.
These are the questions which naturally arise, and the difficulty of solution is largely due to the fact that
The problem is that the treatises on mechanics do not clearly distinguish between:
- mathematical reasoning
- There is no absolute space.
We only conceive of relative motion; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred.
- There is no absolute time.
It is meaningless to say that 2 periods are equal. Its meaning is only by convention.
- We do not even have a direct intuition of the simultaneity of two events occurring in two different places.
I have explained this in “Mesure du Temps.”
- Euclidean geometry is just a convention.
Mechanical facts might be enunciated with reference to a non-Euclidean space which would be less convenient but quite as legitimate as our ordinary space The enunciation would become more complicated, but it still would be possible.
Thus, absolute space, absolute time, and even geometry are not conditions which are imposed on mechanics.
All these things no more existed before mechanics than the French language can be logically said to have existed before the truths which are expressed in French.
We might endeavour to enunciate the fundamental law of mechanics in a language independent of all these conventions.
We should in this way get a clearer idea of those laws in themselves. This is what M. An- drade has tried to do, to some extent at any rate, in his Leçons de Mécanique physique.
Of course the enunciation of these laws would become much more complicated, because all these conventions have been adopted for the very purpose of abbreviating and simplifying the enunciation.
Provisionally, then, we shall admit absolute time and Euclidean geometry.
The Principle of Inertia
A body under the action of no force can only move uniformly in a straight line. Is this a truth imposed on the mind à priori ?
If this be so, how is it that the Greeks ignored it? How could they have believed that motion ceases with the cause of motion?
or, again, that every body, if there is nothing to prevent it, will move in a circle, the noblest of all forms of motion?
If it be said that the velocity of a body cannot change, if there is no reason for it to change, may we not just as legitimately maintain that the position of a body cannot change, or that the curvature of its path cannot change, without the agency of an external cause? Is, then, the principle of inertia, which is not an à priori truth, an experimental fact?
Have there ever been experiments on bodies acted on by no forces? and, if so, how did we know that no forces were acting?
The usual instance is that of a ball rolling for a very long time on a marble table; but why do we say it is under the action of no force? Is it be- cause it is too remote from all other bodies to experience any sensible action?
It is not further from the earth than if it were thrown freely into the air; and we all know that in that case it would be subject to the attraction of the earth.
Teachers of mechanics usually pass rapidly over the example of the ball, but they add that the principle of inertia is verified indirectly by its consequences. This is very badly expressed; they evidently mean that various consequences may be verified by a more general principle, of which the principle of inertia is only a particular case.
I shall propose for this general principle the following enunciation:—The acceleration of a body depends only on its position and that of neighbouring bodies, and on their velocities. Mathematicians would say that the movements of all the material molecules of the universe depend on differential equations of the second order. To the classical mechanics. make it clear that this is really a generalisation of the law of inertia we may again have recourse to our imagination.
The law of inertia, as I have said above, is not imposed on us à priori ; other laws would be just as compatible with the principle of sufficient reason. If a body is not acted upon by a force, instead of supposing that its velocity is unchanged we may suppose that its position or its acceleration is unchanged.
Let us for a moment suppose that one of these two laws is a law of nature, and substitute it for the law of inertia: what will be the natural generalisation? A moment’s reflection will show us. In the first case, we may suppose that the velocity of a body depends only on its position and that of neighbouring bodies; in the second case, that the variation of the acceleration of a body depends only on the position of the body and of neighbouring bodies, on their velocities and accelerations; or, in mathematical terms, the differential equations of the motion would be of the first order in the first case and of the third order in the second.
Suppose a world analogous to our solar system, but one in which by a singular chance the orbits of all the planets have neither eccentricity nor inclination; and further, I supposescience and hypothesis
that the masses of the planets are too small for their mutual perturbations to be sensible. Astronomers living in one of these planets would not hesitate to conclude that the orbit of a star can only be circular and parallel to a certain plane; the position of a star at a given mo- ment would then be sufficient to determine its velocity and path. The law of inertia which they would adopt would be the former of the two hypothetical laws I have mentioned.
Imagine this system to be some day crossed by a body of vast mass and immense velocity coming from distant constellations. All the orbits would be profoundly disturbed. Our astronomers would not be greatly astonished.
They would guess that this new star is in itself quite capable of doing all the mischief; but, they would say, as soon as it has passed by, order will again be established. No doubt the distances of the planets from the sun will not be the same as before the cataclysm, but the orbits will become circular again as soon as the disturbing cause has disappeared.
It would be only when the perturbing body is remote, and when the orbits, instead of being circular are found to be elliptical, that the astronomers would find out their mistake, and discover the necessity of reconstructing their mechanics.the classical mechanics.
We can clearly understand our generalised law of inertia only by opposing it to a contrary hypothesis.
Has this generalised law of inertia been verified by experiment, and can it be so verified?
When Newton wrote the Principia, he certainly regarded this truth as experimentally acquired and demonstrated. It was so in his eyes, not only from the anthropomorphic conception to which I shall later refer, but also because of the work of Galileo.
It was so proved by the laws of Kepler. According to those laws, in fact, the path of a planet is entirely determined by its initial position and initial velocity; this, indeed, is what our generalised law of inertia requires.
For this principle to be only true in appearance—lest we should fear that some day it must be replaced by one of the analogous principles which I opposed to it just now—we must have been led astray by some amazing chance such as that which had led into error our imaginary astronomers.
Such an hypothesis is so unlikely that it need not delay us. No one will believe that there can be such chances; no doubt the probability that two eccentricities are both exactly zero is not smaller than the probability that one is 0.1 and the other 0.2. The probability of a simple event is not smaller than that of a complex one.
If, however, the former does occur, we shall not attribute its occurrence to chance; we shall not be inclined to believe that nature has done it deliberately to deceive us. The hypothesis of an error of this kind being discarded, we may admit that so far as astronomy is concerned our law has been verified by experiment.
But Astronomy is not the whole of Physics. A new experiment might falsify the law in some domain of physics.
An experimental law is always subject to revision. We may always expect to see it replaced by some other and more exact law.
But no one seriously thinks that the law of which we speak will ever be abandoned or amended. This is because it will never be submitted to a decisive test.
In the first place, for this test to be complete, all the bodies of the universe must return with their initial velocities to their initial positions after a certain time. We ought then to find that they would resume their original paths.
But this test is impossible; it can be only partially applied, and even when it is applied there will still be some bodies which will not return to their original positions.
Thus there will be a ready explanation of any breaking down of the law.the classical mechanics.
In Astronomy we see the bodies whose motion we are studying, and in most cases we grant that they are not subject to the action of other invisible bodies. Under these conditions, our law must certainly be either verified or not.
But it is not so in Physics.
If physical phenomena are due to motion, it is to the motion of molecules which we cannot see. If, then, the acceleration of bodies we cannot see depends on something else than the positions or velocities of other visible bodies or of invisible molecules, the existence of which we have been led previously to admit, there is nothing to prevent us from supposing that this something else is the position or velocity of other molecules of which we have not so far suspected the existence.
The law will be safeguarded. Let me express the same thought in another form in mathematical language.
Suppose we are observing n molecules, and find that their 3n co-ordinates satisfy a system of 3n differential equations of the fourth order (and not of the second, as required by the law of inertia). We know that by introducing 3n variable auxiliaries, a system of 3n equations of the fourth order may be reduced to a system of 6n equations of the second order.
If, then, we suppose that the 3n auxiliary variables represent the co-ordinates of n invisible molecules, the result is again conformable to the law of inertia.
To sum up, this law, verified experimentally in some particular cases, may be extended fearlessly to the most general cases; for we know that in these general cases it can nei- ther be confirmed nor contradicted by experiment.
The Law of Acceleration
The acceleration of a body is equal to the force which acts on it divided by its mass. Can this law be verified by experiment?
If so, we have to measure the three magnitudes mentioned in the enunciation: acceleration, force, and mass. I admit that acceleration may be measured, because I pass over the difficulty arising from the measurement of time.
But how are we to measure force and mass?
We do not even know what they are.
What is mass?
Newton replies: “The product of the volume and the density.” “It were better to say,” answer Thomson and Tait, “that density is the quotient of the mass by the volume.”
What is force?
“It is,” replies Lagrange, “that which moves or tends to move a body.” “It is,” according to Kirchoff, “the product of the mass and the acceleration.” Then why not say that mass is the quotient of the force by the acceleration? These difficulties are insurmountable.
When we say force is the cause of motion, we are talking metaphysics; and this definition, if we had to bethe classical mechanics.
content with it, would be absolutely fruitless, would lead to absolutely nothing. For a definition to be of any use it must tell us how to measure force; and that is quite suf- ficient, for it is by no means necessary to tell what force is in itself, nor whether it is the cause or the effect of mo- tion. We must therefore first define what is meant by the equality of two forces.
When are two forces equal?
We are told that it is when they give the same acceleration to the same mass, or when acting in opposite directions they are in equilibrium. This definition is a sham. A force applied to a body cannot be uncoupled and applied to another body as an engine is uncoupled from one train and coupled to another.
It is therefore impossible to say what acceleration such a force, applied to such a body, would give to another body if it were applied to it. It is impossible to tell how two forces which are not acting in exactly opposite directions would behave if they were acting in opposite directions.
It is this definition which we try to materialise, as it were, when we measure a force with a dynamometer or with a balance. Two forces, F and F 0 , which I suppose, for simplicity, to be acting vertically upwards, are respectively applied to two bodies, C and C 0 .
I attach a body weighing P first to C and then to C 0 ; if there is equilibrium in both cases I conclude that the two forces F and F 0 are equal, for they are both equal to the weight of the body P.
But am I certain that the body P has kept its weight when I transferred it from the first body to the second? Far from it. I am certain of the contrary.
I know that the magnitude of the weight varies from one point to another, and that it is greater, for instance, at the pole than at the equator. No doubt the difference is very small, and we neglect it in practice; but a definition must have mathematical rigour;
This rigour does not exist. What I say of weight would apply equally to the force of the spring of a dynamometer, which would vary according to temperature and many other circumstances. Nor is this all. We cannot say that the weight of the body P is applied to the body C and keeps in equilibrium the force F.
What is applied to the body C is the action of the body P on the body C.
On the other hand, the body P is acted on by its weight, and by the reaction R of the body C on P the forces F and A are equal, because they are in equilibrium; the forces A and R are equal by virtue of the principle of action and reaction; and finally, the force R and the weight P are equal because they are in equilibrium.
From these three equalities we deduce the equality of the weight P and the force F.the classical mechanics.
Thus we are compelled to bring into our definition of the equality of two forces the principle of the equality of action and reaction; hence this principle can no longer be regarded as an experimental law but only as a definition.
To recognise the equality of two forces we are then in possession of two rules: the equality of two forces in equilibrium and the equality of action and reaction. But, as we have seen, these are not sufficient, and we are compelled to have recourse to a third rule, and to admit that certain forces—the weight of a body, for instance—are constant in magnitude and direction. But this third rule is an experimental law.
It is only approximately true: it is a bad definition. We are therefore reduced to Kir- choff’s definition: force is the product of the mass and the acceleration. This law of Newton in its turn ceases to be regarded as an experimental law, it is now only a definition.
But as a definition it is insufficient, for we do not know what mass is. It enables us, no doubt, to calculate the ratio of two forces applied at different times to the same body, but it tells us nothing about the ratio of two forces applied to two different bodies.
To fill up the gap we must have recourse to Newton’s third law, the equality of action and reaction, still regarded not as an experimental law but as a definition. Two bodies,science and hypothesis A and B, act on each other; the acceleration of A, multiplied by the mass of A, is equal to the action of B on A; in the same way the acceleration of B, multiplied by the mass of B is equal to the reaction of A on B.
As, by definition, the action and the reaction are equal, the masses of A and B arc respectively in the inverse ratio of their masses. Thus is the ratio of the two masses defined, and it is for experiment to verify that the ratio is constant.
This would do very well if the two bodies were alone and could be abstracted from the action of the rest of the world; but this is by no means the case. The acceleration of A is not solely due to the action of B, but to that of a multitude of other bodies, C, D, . . . .
To apply the preceding rule we must decompose the acceleration of A into many components, and find out which of these components is due to the action of B.
The decomposition would still be possible if we suppose that the action of C on A is simply added to that of B on A, and that the presence of the body C does not in any way modify the action of B on A, or that the presence of B does not modify the action of C on A; that is, if we admit that any two bodies attract each other, that their mutual action is along their join, and is only dependent on their distance apart; if, in a word, we admit the hypothesis of centralthe classical mechanics.
We know that to determine the masses of the heavenly bodies we adopt quite a different principle.
The law of gravitation teaches us that the attraction of two bodies is proportional to their masses; if r is their distance apart, m and m 0 their masses, k a constant, then their attraction will be kmm 0 /r 2.
What we are measuring is therefore not mass, the ratio of the force to the acceleration, but the attracting mass; not the inertia of the body, but its attracting power.
It is an indirect process, the use of which is not indispensable theoretically. We might have said that the attraction is inversely proportional to the square of the distance, without being proportional to the product of the masses, that it is equal to f /r 2 but without having f = kmm 0.
If it were so, we should nevertheless, by observing the relative motion of the celestial bodies, be able to calculate the masses of these bodies.
But have we any right to admit the hypothesis of central forces?
Is this hypothesis rigorously accurate?
Is it certain that it will never be falsified by experiment?
Who will venture to make such an assertion?
And if we must abandon this hypothesis, the building which has been so laboriously erected must fall to the ground.
We have no longer any right to speak of the component of the acceleration of A which is due to the action of B.
We have no means of distinguishing it from that which is due to the action of C or of any other body. The rule becomes inapplicable in the measurement of masses.
What then is left of the principle of the equality of action and reaction?
If we reject the hypothesis of central forces this principle must go too; the geometrical resultant of all the forces applied to the different bodies of a system abstracted from all external action will be zero.
In other words, the motion of the centre of gravity of this system will be uniform and in a straight line. Here would seem to be a means of defining mass.
The position of the centre of gravity evidently depends on the values given to the masses; we must select these values so that the motion of the centre of gravity is uniform and rectilinear.
This will always be possible if Newton’s third law holds good, and it will be in general possible only in one way. But no system exists which is abstracted from all external action; every part of the universe is subject, more or less, to the action of the other parts. The law of the motion of the centre of gravity is only rigorously true when applied to the whole universe.
But then, to obtain the values of the masses we must find the motion of the centre of gravity of the universe.the classical mechanics.
The absurdity of this conclusion is obvious; the motion of the centre of gravity of the universe will be for ever to us unknown.
Nothing, therefore, is left, and our efforts are fruitless.
There is no escape from the following definition, which is only a confession of failure: Masses are co-efficients which it is found convenient to introduce into calculations.
We could reconstruct our mechanics by giving to our masses different values. The new mechanics would be in contradiction neither with experiment nor with the general principles of dynamics (the principle of inertia, proportionality of masses and accelerations, equality of action and reaction, uniform motion of the centre of gravity in a straight line, and areas).
But the equations of this mechanics would not be so simple.
Only the first terms would be less simple—i.e., those we already know through experiment; perhaps the small masses could be slightly altered without the complete equations gaining or losing in simplicity.
Hertz has inquired if the principles of mechanics are rigorously true. “In the opinion of many physicists it seems inconceivable that experiment will ever alter the impregnable principles of mechanics; and yet, what is due to experiment may always be rectified by experiment.”
From what we have just seen these fears would appear to be groundless. The principles of dynamics appeared to us first as experimental truths, but we have been compelled to use them as definitions.
It is by definition that force is equal to the product of the mass and the acceleration; this is a principle which is henceforth beyond the reach of any future experiment.
Thus, action and reaction are equal and opposite.
But these unverifiable principles are absolutely devoid of any significance. They cannot be disproved by experiment, but we can learn from them nothing of any use to us; what then is the use of studying dynamics?
This somewhat rapid condemnation would be rather unfair. There is not in Nature any system perfectly isolated, perfectly abstracted from all external action; but there are systems which are nearly isolated.
If we observe such a system, we can study not only the relative motion of its different parts with respect to each other, but the motion of its centre of gravity with respect to the other parts of the universe.
We then find that the motion of its centre of gravity is nearly uniform and rec- tilinear in conformity with Newton’s Third Law.
This is an experimental fact, which cannot be invalidated by athe classical mechanics.
more accurate experiment. What, in fact, would a more accurate experiment teach us?
It would teach us that the law is only approximately true, and we know that already.
Thus is explained how experiment may serve as a basis for the principles of mechanics, and yet will never invalidate them.
Kirchoff has only followed the general tendency of mathematicians towards nominalism; from this his skill as a physicist has not saved him.
He wanted a definition of a force, and he took the first that came handy; but we do not require a definition of force; the idea of force is primitive, irreducible, indefinable; we all know what it is; of it we have direct intuition.
This direct intuition arises from the idea of effort which is familiar to us from childhood. But in the first place, even if this direct intuition made known to us the real nature of force in itself, it would prove to be an insufficient basis for mechanics;
It would, moreover, be quite useless. The important thing is not to know what force is, but how to measure it.
Everything which does not teach us how to measure it is as useless to the mechanician as, for instance, the subjective idea of heat and cold to the student of heat.
This subjective idea cannot be translated into numbers,science and hypothesis and is therefore useless; a scientist whose skin is an absolutely bad conductor of heat, and who, therefore, has never felt the sensation of heat or cold, would read a thermometer in just the same way as any one else, and would have enough material to construct the whole of the theory of heat.
Now this immediate notion of effort is of no use to us in the measurement of force. It is clear, for example, that I shall experience more fatigue in lifting a weight of 100 lb. than a man who is accustomed to lifting heavy burdens.
But there is more than this. This notion of effort does not teach us the nature of force; it is definitively reduced to a recollection of muscular sensations, and no one will maintain that the sun experiences a muscular sensation when it attracts the earth.
All that we can expect to find from it is a symbol, less precise and less convenient than the arrows (to denote direction) used by geometers, and quite as remote from reality.
Anthropomorphism plays a considerable historic rôle in the genesis of mechanics; perhaps it may yet furnish us with a symbol which some minds may find convenient; but it can be the foundation of nothing of a really scientific or philosophical character.
The Thread School.—M. Andrade, in his Leçons dethe classical mechanics.
Mécanique physique, has modernised anthropomorphic mechanics. To the school of mechanics with which Kirchoff is identified, he opposes a school which is quaintly called the “Thread School.”
This school tries to reduce everything to the consideration of certain material systems of negligible mass, regarded in a state of tension and capable of transmitting considerable effort to distant bodies—systems of which the ideal type is the fine string, wire, or thread.
A thread which transmits any force is slightly lengthened in the direction of that force; the direction of the thread tells us the direction of the force, and the magnitude of the force is measured by the lengthening of the thread.
We may imagine such an experiment as the following:
A body A is attached to a thread; at the other extremity of the thread acts a force which is made to vary until the length of the thread is increased by α, and the acceleration of the body A is recorded. A is then detached, and a body B is attached to the same thread, and the same or another force is made to act until the increment of length again is α, and the acceleration of B is noted.
The experiment is then renewed with both A and B until the increment of length is β. The four accelerations observed should be proportional.science and hypothesis
Here we have an experimental verification of the law of acceleration enunciated above. Again, we may consider a body under the action of several threads in equal tension, and by experiment we determine the direction of those threads when the body is in equilibrium. This is an experimental verification of the law of the composition of forces.
But, as a matter of fact, what have we done?
We have defined the force acting on the string by the deformation of the thread, which is reasonable enough; we have then assumed that if a body is attached to this thread, the effort which is transmitted to it by the thread is equal to the action exercised by the body on the thread; in fact, we have used the principle of action and reaction by considering it, not as an experimental truth, but as the very definition of force.
This definition is quite as conventional as that of Kirchoff, but it is much less general. All the forces are not transmitted by the thread (and to compare them they would all have to be transmitted by identical threads).
If we even admitted that the earth is attached to the sun by an invisible thread, at any rate it will be agreed that we have no means of measuring the increment of the thread.
Nine times out of ten, in consequence, our definition will be in default; no sensethe classical mechanics.
of any kind can be attached to it, and we must fall back on that of Kirchoff. Why then go on in this roundabout way?
You admit a certain definition of force which has a meaning only in certain particular cases. In those cases you verify by experiment that it leads to the law of acceleration.
On the strength of these experiments you then take the law of acceleration as a definition of force in all the other cases.
Would it not be simpler to consider the law of acceleration as a definition in all cases, and to regard the experiments in question, not as verifications of that law, but as verifications of the principle of action and reaction, or as proving the deformations of an elastic body depend only on the forces acting on that body?
Without taking into account the fact that the conditions in which your definition could be accepted can only be very imperfectly fulfilled, that a thread is never without mass, that it is never isolated from all other forces than the reaction of the bodies attached to its extremities.
The ideas expounded by M. Andrade are none the less very interesting. If they do not satisfy our logical requirements, they give us a better view of the historical genesis of the fundamental ideas of mechanics.
The reflections they suggest show us how the human mind passed fromscience and hypothesis a naïve anthropomorphism to the present conception of science.
We see that we end with an experiment which is very particular, and as a matter of fact very crude, and we start with a perfectly general law, perfectly precise, the truth of which we regard as absolute. We have, so to speak, freely conferred this certainty on it by looking upon it as a convention.
Are the laws of acceleration and of the composition of forces only arbitrary conventions? Conventions, yes; arbitrary, no—they would be so if we lost sight of the experiments which led the founders of the science to adopt them, and which, imperfect as they were, were sufficient to justify their adoption.
It is well from time to time to let our attention dwell on the experimental origin of these conventions.