The Classical MechanicsJanuary 28, 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.