Reference: Evolution of Physics
This paper presents Chapter I, section 4 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).
The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding. Feedback on these comments is appreciated.
The heading below is linked to the original materials.
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The Riddle of Motion
So long as we deal only with motion along a straight line, we are far from understanding the motions observed in nature. We must consider motions along curved paths, and our next step is to determine the laws governing such motions. This is no easy task. In the case of rectilinear motion our concepts of velocity, change of velocity, and force proved most useful. But we do not immediately see how we can apply them to motion along a curved path. It is indeed possible to imagine that the old concepts are unsuited to the description of general motion, and that new ones must be created. Should we try to follow our old path, or seek a new one?
We must consider motions along curved paths because that is the reality.
The generalization of a concept is a process very often used in science. A method of generalization is not uniquely determined, for there are usually numerous ways of carrying it out. One requirement, however, must be rigorously satisfied: any generalized concept must reduce to the original one when the original conditions are fulfilled.
Laws governing motion along curved paths must reduce to laws governing motion along a straight line when the path is straight.
We can best explain this by the example with which we are now dealing. We can try to generalize the old concepts of velocity, change of velocity, and force for the case of motion along a curved path. Technically, when speaking of curves, we include straight lines. The straight line is a special and trivial example of a curve. If, therefore, velocity, change in velocity, and force are introduced for motion along a curved line, then they are automatically introduced for motion along a straight line. But this result must not contradict those results previously obtained. If the curve becomes a straight line, all the generalized concepts must reduce to the familiar ones describing rectilinear motion. But this restriction is not sufficient to determine the generalization uniquely. It leaves open many possibilities. The history of science shows that the simplest generalizations sometimes prove successful and sometimes not. We must first make a guess. In our case it is a simple matter to guess the right method of generalization. The new concepts prove very successful and help us to understand the motion of a thrown stone as well as that of the planets.
There are many generalizations possible. Start with a simple generalization. There is a good chance of success.
And now just what do the words velocity, change in velocity, and force mean in the general case of motion along a curved line? Let us begin with velocity. Along the curve a very small body is moving from left to right. Such a small body is often called a particle. The dot on the curve in our drawing shows the position of the particle at some instant of time. What is the velocity corresponding to this time and position? Again Galileo’s clue hints at a way of introducing the velocity. We must, once more, use our imagination and think about an idealized experiment. The particle moves along the curve, from left to right, under the influence of external forces. Imagine that at a given time, and at the point indicated by the dot, all these forces suddenly cease to act. Then, the motion must, according to the law of inertia, be uniform. In practice we can, of course, never completely free a body from all external influences. We can only surmise “what would happen if. . . ?” and judge the pertinence of our guess by the conclusions which can be drawn from it and by their agreement with experiment.
The vector in the next drawing indicates the guessed direction of the uniform motion if all external forces were to vanish. It is the direction of the so-called tangent. Looking at a moving particle through a microscope one sees a very small part of the curve, which appears as a small segment. The tangent is its prolongation. Thus the vector drawn represents the velocity at a given instant. The velocity vector lies on the tangent. Its length represents the magnitude of the velocity, or the speed as indicated, for instance, by the speedometer of a car.
First we guess the direction as the direction of the tangent.
Our idealized experiment about destroying the motion in order to find the velocity vector must not be taken too seriously. It only helps us to understand what we should call the velocity vector and enables us to determine it for a given instant at a given point.
In the next drawing, the velocity vectors for three different positions of a particle moving along a curve are shown. In this case not only the direction but the magnitude of the velocity, as indicated by the length of the vector, varies during the motion.
Direction varies along a curved path, but the speed may vary too, and we indicate that by the length of the vector.
Does this new concept of velocity satisfy the requirement formulated for all generalizations? That is: does it reduce to the familiar concept if the curve becomes a straight line? Obviously it does. The tangent to a straight line is the line itself. The velocity vector lies in the line of the motion, just as in the case of the moving cart or the rolling spheres.
The next step is the introduction of the change in velocity of a particle moving along a curve. This also may be done in various ways, from which we choose the simplest and most convenient. The last drawing showed several velocity vectors representing the motion at various points along the path. The first two of these may be drawn again so that they have a common starting-point, as we have seen is possible with vectors. The dotted vector we call the change in velocity. Its starting-point is the end of the first vector and its endpoint the end of the second vector. This definition of the change in velocity may, at first sight, seem artificial and meaningless. It becomes much clearer in the special case in which vectors (1) and (2) have the same direction. This, of course, means going over to the case of straight-line motion. If both vectors have the same initial point, the dotted vector again connects their endpoints. The drawing is now identical with that on p. 18, and the previous concept is regained as a special case of the new one. We may remark that we had to separate the two lines in our drawing, since otherwise they would coincide and be indistinguishable.
Next we determine the change in velocity, by connecting the heads of the vectors by a dotted line.
We now have to take the last step in our process of generalization. It is the most important of all the guesses we have had to make so far. The connection between force and change in velocity has to be established so that we can formulate the clue which will enable us to understand the general problem of motion.
The clue to an explanation of motion along a straight line was simple: external force is responsible for change in velocity; the force vector has the same direction as the change. And now what is to be regarded as the clue to curvilinear motion? Exactly the same! The only difference is that change of velocity has now a broader meaning than before. A glance at the dotted vectors of the last two drawings shows this point clearly. If the velocity is known for all points along the curve, the direction of the force at any point can be deduced at once. One must draw the velocity vectors for two instants separated by a very short time interval and therefore corresponding to positions very near each other. The vector from the end-point of the first to that of the second indicates the direction of the acting force. But it is essential that the two velocity vectors should be separated only by a “very short” time interval. The rigorous analysis of such words as “very near”, “very short” is far from simple. Indeed it was this analysis which led Newton and Leibnitz to the discovery of differential calculus.
Next we determine the force responsible for the change in velocity.
It is a tedious and elaborate path which leads to the generalization of Galileo’s clue. We cannot show here how abundant and fruitful the consequences of this generalization have proved. Its application leads to simple and convincing explanations of many facts previously incoherent and misunderstood.
From the extremely rich variety of motions we shall take only the simplest and apply to their explanation the law just formulated.
A bullet shot from a gun, a stone thrown at an angle, a stream of water emerging from a hose, all describe familiar paths of the same type the parabola. Imagine a speedometer attached to a stone, for example, so that its velocity vector may be drawn for any instant. The result may well be that represented in the above drawing. The direction of the force acting on the stone is just that of the change in velocity, and we have seen how it may be determined. The result, shown in the next drawing, indicates that the force is vertical, and directed downward. It is exactly the same as when a stone is allowed to fall from the top of a tower. The paths are quite different, as also are the velocities, but the change in velocity has the same direction, that is, toward the centre of the earth.
This generalization provides clear explanation of many observed motions.
A stone attached to the end of a string and swung around in a horizontal plane moves in a circular path.
All the vectors in the diagram representing this motion have the same length if the speed is uniform. Nevertheless, the velocity is not uniform, for the path is not a straight line. Only in uniform, rectilinear motion are there no forces involved. Here, however, there are, and the velocity changes not in magnitude but in direction. According to the law of motion there must be some force responsible for this change, a force in this case between the stone and the hand holding the string. A further question arises immediately: in what direction does the force act? Again a vector diagram shows the answer. The velocity vectors for two very near points are drawn, and the change of velocity found. This last vector is seen to be directed along the string toward the centre of the circle, and is always perpendicular to the velocity vector, or tangent. In other words, the hand exerts a force on the stone by means of the string.
The vector diagram shows the direction of the force.
Very similar is the more important example of the revolution of the moon around the earth. This may be represented approximately as uniform circular motion. The force is directed toward the earth for the same reason that it was directed toward the hand in our former example. There is no string connecting the earth and the moon, but we can imagine a line between the centres of the two bodies; the force lies along this line and is directed toward the centre of the earth, just as the force on a stone thrown in the air or dropped from a tower.
When a body is revolving around another the force lies along the line joining their centers.
All that we have said concerning motion can be summed up in a single sentence. Force and change of velocity are vectors having the same direction. This is the initial clue to the problem of motion, but it certainly does not suffice for a thorough explanation of all motions observed. The transition from Aristotle’s line of thought to that of Galileo formed a most important corner-stone in the foundation of science. Once this break was made, the line of further development was clear. Our interest here lies in the first stages of development, in following initial clues, in showing how new physical concepts are born in the painful struggle with old ideas. We are concerned only with pioneer work in science, which consists of finding new and unexpected paths of development; with the adventures in scientific thought which create an ever-changing picture of the universe. The initial and fundamental steps are always of a revolutionary character. Scientific imagination finds old concepts too confining, and replaces them by new ones. The continued development along any line already initiated is more in the nature of evolution, until the next turning point is reached when a still newer field must be conquered. In order to understand, however, what reasons and what difficulties force a change in important concepts, we must know not only the initial clues, but also the conclusions which can be drawn.
The above sums up to: Force lies in the same direction as the change of velocity. Galileo did the pioneering work here. The initial and fundamental steps are always of a revolutionary character.
One of the most important characteristics of modern physics is that the conclusions drawn from initial clues are not only qualitative but also quantitative. Let us again consider a stone dropped from a tower. We have seen that its velocity increases as it falls, but we should like to know much more. Just how great is this change? And what is the position and the velocity of the stone at any time after it begins to fall? We wish to be able to predict events and to determine by experiment whether observation confirms these predictions and thus the initial assumptions.
In modern physics the conclusions that are drawn from initial clues are not only qualitative but also quantitative, so they can be verified experimentally.
To draw quantitative conclusions we must use the language of mathematics. Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone. To follow up these ideas demands the knowledge of a highly refined technique of investigation. Mathematics as a tool of reasoning is necessary if we wish to draw conclusions which may be compared with experiment. So long as we are concerned only with fundamental physical ideas we may avoid the language of mathematics. Since in these pages we do this consistently, we must occasionally restrict ourselves to quoting, without proof, some of the results necessary for an understanding of important clues arising in the further development. The price which must be paid for abandoning the language of mathematics is a loss in precision, and the necessity of sometimes quoting results without showing how they were reached.
So long as we are concerned only with fundamental physical ideas we may avoid the language of mathematics. But mathematics as a tool of reasoning is necessary if we wish to draw conclusions which may be compared with experiment.
A very important example of motion is that of the earth around the sun. It is known that the path is a closed curve, called the ellipse. The construction of a vector diagram of the change in velocity shows that the force on the earth is directed toward the sun. But this, after all, is scant information. We should like to be able to predict the position of the earth and the other planets for any arbitrary instant of time, we should like to predict the date and duration of the next solar eclipse and many other astronomical events. It is possible to do these things, but not on the basis of our initial clue alone, for it is now necessary to know not only the direction of the force but also its absolute value its magnitude. It was Newton who made the inspired guess on this point. According to his law of gravitation the force of attraction between two bodies depends in a simple way on their distance from each other. It becomes smaller when the distance increases. To be specific it becomes 2×2=4 times smaller if the distance is doubled, 3×3 = 9 times smaller if the distance is made three times as great.
Moon around the Earth and Earth around the Sun are examples of motions from which a lot can be learned. Newton made an inspired guess to compute the force necessary for such motion.
Thus we see that in the case of gravitational force we have succeeded in expressing, in a simple way, the dependence of the force on the distance between the moving bodies. We proceed similarly in all other cases where forces of different kinds for instance, electric, magnetic, and the like are acting. We try to use a simple expression for the force. Such an expression is justified only when the conclusions drawn from it are confirmed by experiment.
But this knowledge of the gravitational force alone is not sufficient for a description of the motion of the planets. We have seen that vectors representing force and change in velocity for any short interval of time have the same direction, but we must follow Newton one step farther and assume a simple relation between their lengths. Given all other conditions the same, that is, the same moving body and changes considered over equal time intervals, then, according to Newton, the change of velocity is proportional to the force.
Newton went one step farther and found that the change of velocity is proportional to the force.
Thus just two complementary guesses are needed for quantitative conclusions concerning the motion of the planets. One is of a general character, stating the connection between force and change in velocity. The other is special, and states the exact dependence of the particular kind of force involved on the distance between the bodies. The first is Newton’s general law of motion, the second his law of gravitation. Together they determine the motion. This can be made clear by the following somewhat clumsy-sounding reasoning. Suppose that at a given time the position and velocity of a planet can be determined, and that the force is known. Then, according to Newton’s laws, we know the change in velocity during a short time interval. Knowing the initial velocity and its change, we can find the velocity and position of the planet at the end of the time interval. By a continued repetition of this process the whole path of the motion may be traced without further recourse to observational data. This is, in principle, the way mechanics predicts the course of a body in motion, but the method used here is hardly practical. In practice such a step-by-step procedure would be extremely tedious as well as inaccurate. Fortunately, it is quite unnecessary; mathematics furnishes a short cut, and makes possible precise description of the motion in much less ink than we use for a single sentence. The conclusions reached in this way can be proved or disproved by observation.
Newton’s discoveries and mathematics allows us to easily determine the path of the planets.
The same kind of external force is recognized in the motion of a stone falling through the air and in the revolution of the moon in its orbit, namely, that of the earth’s attraction for material bodies. Newton recognized that the motions of falling stones, of the moon, and of planets are only very special manifestations of a universal gravitational force acting between any two bodies. In simple cases the motion may be described and predicted by the aid of mathematics. In remote and extremely complicated cases, involving the action of many bodies on each other, a mathematical description is not so simple, but the fundamental principles are the same.
The principle of universal gravitation, discovered by Newton is simple but the mathematical description of cases where many bodies are involved is not so simple.
We find the conclusions, at which we arrived by following our initial clues, realized in the motion of a thrown stone, in the motion of the moon, the earth, and the planets.
It is really our whole system of guesses which is to be either proved or disproved by experiment. No one of the assumptions can be isolated for separate testing. In the case of the planets moving around the sun it is found that the system of mechanics works splendidly. Nevertheless we can well imagine that another system, based on different assumptions, might work just as well.
There may be another system based on different assumptions that may work even better to describe the phenomenon of motion due to gravity.
Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavour to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility or the meaning of such a comparison. But he certainly believes that, as his knowledge increases, his picture of reality will become simpler and simpler and will explain a wider and wider range of his sensuous impressions. He may also believe in the existence of the ideal limit of knowledge and that it is approached by the human mind. He may call this ideal limit the objective truth.
Our picture of reality will grow and become simpler as knowledge increases. We shall then be able to explain wider range of our observations and experiences.
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FINAL COMMENTS
Objective reality is based on the consistency (harmony and continuity) of observations. The greater is the range of consistency, the more knowledge we have to understand our experiences.
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