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Feynman Physics Lectures V1 Ch11 1961-11-03 Vectors

Feynman Physics Lectures V1 Ch11 1961-11-03 Vectors - 11...

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Unformatted text preview: 11 Vectors 11-1 Symmetry in physics In this chapter we introduce a subject that is technically known in physics as symmetry in physical law. The word “symmetry” is used here with a special meaning, and therefore needs to be defined. When is a thing symmetrical—how can we define it? When we have a picture that is symmetrical, one side is somehow the same as the other side. Professor Hermann Weyl has given this definition of symmetry: a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation. For instance, if we look at a vase that is left-and-right symmetrical, then turn it 180° around the vertical axis, it looks the same. We shall adopt the definition of symmetry in Weyl’s more general form, and in that form we shall discuss symmetry of physical laws. Suppose we build a complex machine in a certain place, with a lot of compli- cated interactions, and balls bouncing around with forces between them, and so on. Now suppose we build exactly the same kind of equipment at some other place, matching part by part, with the same dimensions and the same orientation, every- thing the same only displaced laterally by some distance. Then, if we start the two machines in the same initial circumstances, in exact correspondence, we ask: will one machine behave exactly the same as the other? Will it follow all the mo- tions in exact parallelism? Of course the answer may well be no, because if we choose the wrong place for our machine it might be inside a wall and interferences from the wall would make the machine not work. All of our ideas in physics require a certain amount of common sense in their application; they are not purely mathematical or abstract ideas. We have to under- stand what we mean when we say that the phenomena are the same when we move the apparatus to a new position. We mean that we move everything that we believe is relevant; if the phenomenon is not the same, we suggest that something relevant has not been moved, and we proceed to look for it. If we never find it, then we claim that the laws of physics do not have this symmetry. On the other hand, we may find it—we expect to find it—if the laws of physics do have this symmetry; looking around, we may discover, for instance, that the wall is pushing on the apparatus. The basic question is, if we define things well enough, if all the essential forces are included inside the apparatus, if all the relevant parts are moved from one place to another, will the laws be the same? Will the machinery work the same way? It is clear that what we want to do is to move all the equipment and essential influences, but not everything in the world—planets, stars, and all—for if we do that, we have the same phenomenon again for the trivial reason that we are right back where we started. No, we cannot move everything. But it turns out in practice that with a certain amount of intelligence about what to move, the ma- chinery will work. In other words, if we do not go inside a wall, if we know the origin of the outside forces, and arrange that those are moved too, then the ma- chinery will work the same in one location as in another. 11-2 Translations We shall limit our analysis to just mechanics, for which we now have sufficient knowledge. In previous chapters we have seen that the laws of mechanics can be summarized by a set of three equations for each particle: m(d2x/d12) = F,, m(d2y/dt2) = F,, m(d22/dt2) = F,. (11.1) 11—1 11—1 Symmetry in physics 11—2 Translations 11—3 Rotations 11—4 Vectors 11—5 Vector algebra 11—6 Newton’s laws in vector notation 11—7 Scalar product of vectors Fig. 11—1. systems. Two parallel coordinate Now this means that there exists a way to measure x, y, and 2 on three perpendicu- lar axes, and the forces along those directions, such that these laws are true. These must be measured from some origin, but where do we put the origin? All that Newton would tell us at first is that there is some place that we can measure from, perhaps the center of the universe, such that these laws are correct. But we can show immediately that we can never find the center, because if we use some other origin it would make no diflerence. In other words, suppose that there are two people—Joe, who has an origin in one place, and Moe, who has a parallel system whose origin is somewhere else (Fig. 11—1). Now when Joe measures the location of the point in space, he finds it at x, y, and 2 (we shall usually leave 2 out because it is too confusing to draw in a picture). Moe, on the other hand, when measuring the same point, will obtain a dlfferent x (in order to distinguish it, we will call it x’), and in principle a different y, although in our example they are numerically equal. So we have I I x = x — a, y = y, z’ = 2. (11.2) Now in order to complete our analysis we must know what Moe would obtain for the forces. The force is supposed to act along some line, and by the force in the x-direction we mean the part of the total which is 1n the x-direction, which is the magnitude of the force times this cosine of its angle with the x-axis. Now we see that Moe would use exactly the same prOJection as Joe would use, so we have a set of equations F, = F,, F, = F,, F, = F,. (11.3) These would be the relationships between quantities as seen by Joe and Moe. The question is, if Joe knows Newton’s laws, and if Moe tries to write down Newton’s laws, will they also be correct for him? Does it make any difl‘erence from which origin we measure the points? In other words, assuming that equations (11.1) are true, and the Eqs. (11.2) and (11.3) give the relationship of the measure- ments, is it or is it not true that (a) m(d2x’/dtz) = (b) m(d2y’/dt2) = Fy’; (c) m(d22’/dt2) = 12,? (11.4) In order to test these equations we shall differentiate the formula for x’ twice. First of all dx’_d( _ )[email protected] “‘“X a ‘d: dt Now we shall assume that Moe’s origin is fixed (not moving) relative to Joe’s; therefore a is a constant and da/dt = 0, so we find that dx’/dt = dx/dt’ and therefore a’2x’/alt2 = d2x/dt2; therefore we know that Eq. (11.4a) becomes m(d2x/dt2) = F,“ (We also suppose that the masses measured by Joe and Moe are equal.) Thus the acceleration times the mass is the same as the other fellow’s. We have also found the formula for F5, for, substituting from Eq. (11.1), we find that F,» = F1. Therefore the laws as seen by Moe appear the same; he can write Newton’s laws too, with different coordinates, and they will still be right. That means that 11—2 there is no unique way to define the origin of the world, because the laws will appear the same, from whatever position they are observed. This is also true: if there is a piece of equipment in one place with a certain kind of machinery in it, the same equipment in another place will behave in the same way. Why? Because one machine, when analyzed by Moe, has exactly the same equations as the other one, analyzed by Joe. Since the equations are the same, the phenomena appear the same. So the proof that an apparatus in a new position behaves the same as it did in the old position is the same as the proof that the equations when displaced in space reproduce themselves. Therefore we say that the laws of physics are symmetrical for translational displacements, symmetrical in the sense that the laws do not change when we make a translation of our co- ordinates. Of course it is quite obvious intuitively that this is true, but it is inter- esting and entertaining to discuss the mathematics of it. 11—3 Rotations The above is the first of a series of ever more complicated propositions con- cerning the symmetry of a physical law. The next proposition is that it should make no difference in which direction we choose the axes. In other words, if we build a piece of equipment in some place and watch it operate, and nearby we build the same kind of apparatus but put it up on an angle, will it operate in the same way? Obviously it will not if it is a Grandfather clock, for example! If a pendulum clock stands upright, it works fine, but if it is tilted the pendulum falls against the side of the case and nothing happens. The theorem is then false in the case of the pendulum clock, unless we include the earth, which is pulling on the pendulum. Therefore we can make a prediction about pendulum clocks if we believe in the symmetry of physical law for rotation: something else is involved in the operation of a pendulum clock besides the machinery of the clock, something outside it that we should look for. We may also predict that pendulum clocks will not work the same way when located in different places relative to this mysterious source of asymmetry, perhaps the earth. Indeed, we know that a pendulum clock up in an artificial satellite, for example, would not tick either, because there is no effective force, and on Mars it would go at a different rate. Pendulum clocks do involve something more than just the machinery inside, they involve something on the outside. Once we recognize this factor, we see that we must turn the earth along with the apparatus. Of course we do not have to worry about that, it is easy to do; one simply waits a moment or two and the earth turns; then the pendulum clock ticks again in the new position the same as it did before. While we are rotating in space our angles are always changing, absolutely; this change does not seem to bother us very much, for in the new position we seem to be in the same condition as in the old. This has a certain tendency to confuse one, because it is true that in the new turned position the laws are the same as in the unturned position, but it is not true that as we turn a thing it follows the same laws as it does when we are not turning it. If we perform sufficiently delicate experiments, we can tell that the earth is rotating, but not that it had rotated. In other words, we cannot locate its angular position, but we can tell that it is changing. Now we may discuss the effects of angular orientation upon physical laws. Let us find out whether the same game with Joe and Moe works again. This time, to avoid needless complication, we shall suppose that Joe and Moe use the same origin (we have already shown that the axes can be moved by translation to another place). Assume that Moe’s axes have rotated relative to Joe’s by an angle 0. The two coordinate systems are shown in Fig. 11—2, which is restricted to two dimensions. Consider any point P having coordinates (x, y) in Joe’s system and (x’, y’) in Moe’s system. We shall begin, as in the previous case, by expressing the coordinates x’ and y’ in terms of x, y, and 6. To do so, we first drop perpendic- ulars from P to all four axes and draw AB perpendicular to PQ. Inspection of the figure shows that x’ can be written as the sum of two lengths along the x’-axis, and y’ as the difference of two lengths along AB. All these lengths are expressed 11—3 Fig. 11—2. Two coordinate systems having different angular orientations. Fig. 11—3. Components of a force in the two systems. in terms of x, y, and 0 in equations (11.5), to which we have added an equation for the third dimension. x = xcos0 + ysin 0, y = ycos6 — xsin 9, (11.5) 2:2. The next step is to analyze the relationship of forces as seen by the two observers, following the same general method as before. Let us assume that a force F, which has already been analyzed as having components FE and F, (as seen by Joe), is acting on a particle of mass m, located at point P in Fig. 11—2. For simplicity, let us m0ve both sets of axes so that the origin is at P, as shown in Fig. 11—3. Moe sees the components of F along his axes as Fri and F,,:. F, has components along both the x’- and y’-axes, and F1, likewise has components along both these axes. To express F11 in terms of F, and Fy, we sum these components along the x’-axis, and in a like manner we can express Fy: in terms of F: and Fy. The results are F,: = F, cos 0 + F” sin 6, F”: = F, cos 0 — F, sin 6, F,» = F2. (11.6) It is interesting to note an accident of sorts, which is of extreme importance: the formulas (11.5) and (11.6), for coordinates of P and components of F, respectively, are of identical form. As before, Newton’s laws are assumed to be true in Joe’s system, and are expressed by equations (11.1). The question, again, is whether Moe can apply Newton’s laws—will the results be correct for his system of rotated axes? In other words, if we assume that Eqs. (11.5) and (11.6) give the relationship of the measure- ments, is it true or not true that m(d2x’/dt2) = m(d2y’/dt2) = F), m(d22’/dt2) = F2]? (11.7) To test these equations, we calculate the left and right sides independently, and compare the results. To calculate the left sides, we multiply equations (11.5) by m, and differentiate twice with respect to time, assuming the angle 0 to be constant. This gives m(d2x’/a't2) = m(d2x/dt2) cos 0 + m(d2y/dt2) sin 6, m(d2y’/dt2) = m(d2y/d12) cos 0 — m(d2x/dt2) sin 0, m(d2z’/dt2) = m(dzz/dt2). (11.8) We calculate the right sides of equations (11.7) by substituting equations (11.1) into equations (11.6). This gives Fxr = m(d2x/dt2) cos 0 + m(d2y/dt2) sin 0, Fyr = m(d2y/dt2) cos 0 — m(d2x/dt2) sin 0, F, = m(d22/dt2). (11.9) Behold! The right sides of Eqs. (11.8) and (11.9) are identical, so we conclude that if Newton’s laws are correct on one set of axes, they are also valid on any other set of axes. This result, which has now been established for both translation and rotation of axes, has certain consequences: first, no one can claim his particular axes are unique, but of course they can be more convenient for certain particular problems. For example, it is handy to have gravity along one axis, but this is not physically necessary. Second, it means that any piece of equipment which is completely self-contained, with all the force-generating equipment completely in- side the apparatus, would work the same when turned at an angle. 11—4 11—4 Vectors Not only Newton’s laws, but also the other laws of physics, so far as we know today, have the two properties which we call invariance (or symmetry) under translation of axes and rotation of axes. These properties are so important that a mathematical technique has been developed to take advantage of them in writing and using physical laws. The foregoing analysis involved considerable tedious mathematical work. To reduce the details to a minimum in the analysis of such questions, a very power- ful mathematical machinery has been devised. This system, called vector analysis, supplies the title of this chapter; strictly speaking, however, this is a chapter on the symmetry of physical laws. By the methods of the preceding analysis we were able to do everything required for obtaining the results that we sought, but in practice we should like to do things more easily and rapidly, so we employ the vector technique. We began by noting some characteristics of two kinds of quantities that are important in physics. (Actually there are more than two, but let us start out with two.) One of them, like the number of potatoes in a sack, we call an ordinary quantity, or an undirected quantity, or a scalar. Temperature is an example of such a quantity. Other quantities that are important in physics do have direction, for instance velocity: we have to keep track of which way a body is going, not just its speed. Momentum and force also have direction, as does displacement: when someone steps from one place to another in space, we can keep track of how far he went, but if we wish also to know where he went, we have to specify a direction. All quantities that have a direction, like a step in space, are called vectors. A vector is three numbers. In order to represent a step in space, say from the origin to some particular point P whose location is (x, y, 2), we really need three numbers, but we are going to invent a single mathematical symbol, r, which is unlike any other mathematical symbols we have so far used.* It is not a single number, it represents three numbers: x, y, and 2. It means three numbers, but not really only those three numbers, because if we were to use a different coordinate system, the three numbers would be changed to x’, y’, and 2’. However, we want to keep our mathematics simple and so we are going to use the same mark to repre- sent the three numbers (x, y, z) and the three numbers (x’, y’, 2’). That is, we use the same mark to represent the first set of three numbers for one coordinate system, but the second set of three numbers if we are using the other coordinate system. This has the advantage that when we change the coordmate system, we do not have to change the letters of our equations. If we write an equation in terms of x, y, z, and then use another system, we have to change to x’, y’, 2’, but we shall just write 1-, with the convention that it represents (x, y, 2) if we use one set of axes, or (x’, y’, 2’) if we use another set of axes, and so on. The three numbers which describe the quantity in a g1ven coordinate system are called the components of the vector in the direction of the coordinate axes of that system. That is, we use the same symbol for the three letters that correspond to the same object, as seen from different axes. The very fact that we can say “the same object” implies a physical intuition about the reality of a step in space, that is independent of the components in terms of which we measure it. So the symbol r will represent the same thing no matter how we turn the axes. Now suppose there is another directed physical quantity, any other quantity, which also has three numbers associated with it, like force, and these three numbers change to three other numbers by a certain mathematical rule, if we change the axes. It must be the same rule that changes (x, y, 2) into (x’, y’, 2’). In other words, any physical quantity associated with three numbers which transform as do the components of a step in space is a vector. An equation like F=r would thus be true in any coordinate system if it were true in one. This equation, * In type, vectors are represented by boldface; 1n handwritten form an arrow is used :72 11—5 of course, stands for the three equations F2: = x3 F1] = y; Fz = z: or, alternatively, for F2, = x’, F,,: y’, F, = z’. The fact that a physical relationship can be expressed as a vector equation assures us the relationship is unchanged by a mere rotation of the coordinate system. That is the reason why vectors are so useful in physics. Now let us examine some of the properties of vectors. As examples of vectors we may mention velocity, momentum, force, and acceleration. For many purposes it is convenient to represent a vector quantity by an arrow that indicates the direc- tion in which it is acting. Why can we represent force, say, by an arrow? Because it has the same mathematical transformation properties as a “step in space.” We thus represent it in a diagram as if it were a step, using a scale such that one unit of force, or one newton, corresponds to a certain convenient length. Once we have done this, all forces can be represented as lengths, because an equation like F=kr, where k is some constant, is a perfectly legitimate equation. Thus we can always represent forces by lines, which is very convenient, because once we have drawn the line we no longer need the axes. Of course, we can quickly calculate the three components as they change upon turning the axes, because that is just a geometric problem. 11—5 Vector algebra Now we must describe the laws, or rules, for combining vectors in various ways. The first such combination is the addition of two vectors: suppose that a is a vector which in some particular coordinate system has the three components (ax, ay, a,), and that b is another vector which has the three components (bx, by, b,). Now let us invent three new numbers (ax + b,, a, + by, a, + b,). Do these form a vector? “Well,” we might say, “they are three numbers, and every t...
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