As an interesting example, let us consideran elastic collision between two objects of equal mass. If theycome together with the same speed, they would come apart at that same speed, bysymmetry. But now look at this in another circumstance, in which one of them ismoving with velocity v and the other one is at rest. What happens? We have been through thisbefore. We watch the symmetrical collision from a car moving along with one ofthe objects, and we find that if a stationary body is struck elastically byanother body of exactly the same mass, the moving body stops, and the one thatwas standing still now moves away with the same speed that the other one had;the bodies simply exchange velocities. This behavior can easily be demonstratedwith a suitable impact apparatus. More generally, if both bodies are moving,with different velocities, they simply exchange velocity at impact.
作为一种有趣的例子，让我们来看一个碰撞：两个对象，质量相等。如果它们相向而行时，速度一样，那么，由于对称，它们在相背而行时，速度也一样。但是现在，在另外一个情况下，来看这个，这种情况就是，一个以速度v移动，另一个是静止的。会发生什么呢？我们前面讨论过这个。一个卡车，沿着一个对象移动的方向移动，我们坐在卡车中，观察对称性的碰撞，我们发现，如果一个静止的物体，被一个同等质量的物体撞击，那么，前者停止，后者会以前者的速度来移动，物体之间，只是简单地交换了速度。这个表现，用一个合适的撞击装置，可以很容易地演示。更普遍一些，如果两个物体，都以不同的矢量速度移动，那么，它们相撞时，只是简单地交换矢量速度。

Another example of an almost elastic interactionis magnetism. If we arrange a pair of U-shaped magnets in our glide blocks, sothat they repel each other, when one drifts quietly up to the other, it pushesit away and stands perfectly still, and now the other goes along, frictionlessly.
另一个例子，就是磁铁，也几乎是弹性的交互作用。如果在我们的滑动块中，安排一对U-型磁铁，这样，它们就将互斥，当一个静静地飘向另一个时，会把另一个推开，然后自己完全静止，现在，另外一个开始毫无摩擦地走了。

The principle of conservation of momentumis very useful, because it enables us to solve many problems without knowingthe details. We did not know the details of the gas motions in the cap explosion,yet we could predict the velocities with which the bodies came apart, forexample. Another interesting example is rocket propulsion. A rocket of largemass, M , ejects a small piece, of mass m , with a terrific velocity V relative to the rocket. After this the rocket, if it were originallystanding still, will be moving with a small velocity, v . Using the principle of conservation of momentum, we can calculatethis velocity to be
v=(m/M)*V.
So long as material is being ejected, the rocket continues to pick upspeed. Rocket propulsion is essentially the same as the recoil of a gun: thereis no need for any air to push against.
动量守恒原理,非常有用，因为，它可以让我们在不知道细节的前提下,解决很多问题。例如，在帽子装置的爆炸中,我们并不知道气体运动的细节，但尽管如此，我们可以预测物体分开的矢量速度。另一个有趣的例子，就是火箭推进。一个质量为M的火箭，用一个相对于火箭的、极大的速度V，喷射很小的质量m。在此之后，如果火箭原先是静止的，它将会以一个小的速度v移动。利用动量守恒原理，我们可以计算出这个速度为：
v=(m/M)*V
只要材料一直在被喷射，那么，火箭就会一直得到加速。火箭推进，本质上，与枪的后坐力是一样的：不需要任何空气来推进。

10–5Relativistic momentum 10-5 相对论里的动量
In modern times the law of conservation of momentumhas undergone certain modifications.However, the law is still true today, themodifications being mainly in the definitions of things. In the theory ofrelativity it turns out that we do have conservation of momentum; the particleshave mass and the momentum is still given by mv , the mass times the velocity, but the mass changes with thevelocity, hence the momentum also changes. The mass varies with velocityaccording to the law
m=m01−v2/c2−−−−−−−−√,(10.7)
where m0 is the mass of the body at rest and c is the speed of light. It is easy to see from the formula that thereis negligible difference between m and m0 unless v is very large, and that for ordinary velocities the expression formomentum reduces to the old formula.
.在现代，动量守恒规律，已经经历了某些修改。然而，此规律今天依然正确，修改主要是在事物的定义方面。从相对论看，我们没有动量守恒；粒子有质量，动量还是通过质量乘以矢量速度mv被给予，但质量是随矢速而改变的，因此，动量也会改变。依据此规律，质量随着矢速而改变：
(10.7)
这里，m0是物体静止时的质量，c是光速。从此公式，很易看出，除非v 非常大，否则m与 m0之间的差别，可以忽略，所以，对于通常的矢速来说，动量的表达式，还是回到老的公式。

The components of momentum for a singleparticle are written as
px=m0vx1−v2/c2−−−−−−−−√,py=m0vy1−v2/c2−−−−−−−−√,pz=m0vz1−v2/c2−−−−−−−−√,(10.8)
where v2=v2x+v2y+v2z. If the x -components are summed over all the interacting particles, both beforeand after a collision, the sums are equal; that is, momentum is conserved in thex -direction. The same holds true in any direction.
对于单个的粒子，动量的分量可写为：
(10.8)
这里v2=v2x+v2y+v2z。如果x分量被假定为是遍及所有交互着的粒子的，无论是碰撞前还是碰撞后，总和就是相等的；也就是说，在x方向，动量是守恒的。对任何方向，都如此。

In Chapter 4we saw that the law of conservation of energy is not valid unless we recognizethat energy appears in different forms, electrical energy, mechanical energy,radiant energy, heat energy, and so on. In some of these cases, heat energy forexample, the energy might be said to be “hidden.” This example might suggestthe question, “Are there also hidden forms of momentum—perhaps heat momentum?”The answer is that it is very hard to hide momentum for the following reasons.
在第4章，我们曾看到，除非我们能认识到能量以不同形式出现，如电能、机械能、辐射能、热能等，否则的话，能量守恒规律就并不有效。在有些形式中，能量可以说是被“隐藏”了，例如热能。这个例子，可引出问题：“是否也有隐藏形式的动量，比如热动量？”答案是，由于下面的原因，动量很难被隐藏。

The random motions of the atoms of a body furnisha measure of heat energy, if the squares of the velocities are summed. Thissum will be a positive result, having no directional character. The heat isthere, whether or not the body moves as a whole, and conservation of energy inthe form of heat is not very obvious. On the other hand, if one sums the velocities,which have direction, and finds a result that is not zero, that means thatthere is a drift of the entire body in some particular direction, and such agross momentum is readily observed. Thus there is no random internal lostmomentum, because the body has net momentum only when it moves as a whole.Therefore momentum, as a mechanical quantity, is difficult to hide. Nevertheless,momentum can be hidden—in the electromagnetic field, for example. Thiscase is another effect of relativity.
物体原子的随机运动，提供了对热能的测量方法，如果速度的平方被假定的话。{？使用}这个总和，将是正的结果，没有任何方向性的特点。热就在那里，不论物体是否作为一个整体在移动，而以热能形式出现的能量守恒，并不很明显。另一方面，矢速是有方向的，如果有人把矢速加起来，所得结果，非零，那就意味着，整个物体，在某一具体方向，有漂移，于是，这样一种净的动量，已经可被观察了。这样，没有随机的内部损失的动量，因为，只有当物体作为一个整体运动时，它才有净的动量。因此，动量作为一种机械的量，很难隐藏。尽管如此，动量可被隐藏，例如，在电磁领域。这种情况，是相对论的另一种效应。

One of the propositions of Newton was thatinteractions at a distance are instantaneous. It turns out that such is not thecase; in situations involving electrical forces, for instance, if an electricalcharge at one location is suddenly moved, the effects on another charge, at anotherplace, do not appear instantaneously—there is a little delay. In thosecircumstances, even if the forces are equal the momentum will not check out;there will be a short time during which there will be trouble, because for a whilethe first charge will feel a certain reaction force, say, and will pick up somemomentum, but the second charge has felt nothing and has not yet changed itsmomentum. It takes time for the influence to cross the intervening distance,which it does at 186,000 miles a second. In that tiny time the momentum of the particlesis not conserved. Of course after the second charge has felt the effect of thefirst one and all is quieted down, the momentum equation will check out allright, but during that small interval momentum is not conserved. We representthis by saying that during this interval there is another kind of momentumbesides that of the particle, mv , and that is momentum in the electromagnetic field. If we add thefield momentum to the momentum of the particles, then momentum is conserved atany moment all the time. The fact that the electromagnetic field can possessmomentum and energy makes that field very real, and so, for better understanding,the original idea that there are just the forces between particles has to bemodified to the idea that a particle makes a field, and a field acts on anotherparticle, and the field itself has such familiar properties as energy contentand momentum, just as particles can have. To take another example: anelectromagnetic field has waves, which we call light; it turns out that lightalso carries momentum with it, so when light impinges on an object it carriesin a certain amount of momentum per second; this is equivalent to a force,because if the illuminated object is picking up a certain amount of momentumper second, its momentum is changing and the situation is exactly the same asif there were a force on it. Light can exert pressure by bombarding an object;this pressure is very small, but with sufficiently delicate apparatus it ismeasurable.
牛顿的提议之一，就是在一定距离的交互作用，是瞬间的。而结果显示，情况并非如此；例如，在牵扯到电力的情况下，如果某地的一个电荷，突然移动，那么，它对另一地方的另一电荷的影响，并不会立即表现出来--有一个小的延迟。在这些情况中，即便力是相等的，动量也不会结账；将会有一个短的时间，在此期间，会有麻烦，因为，有那么一个片刻，第一个电荷，会感到一定的反作用力，这么说吧，它会积累一些动量，但是，第二个电荷，什么都没感觉到，其动量，尚未改变。要让影响，穿过{相互}干涉的距离，需要时间，大约是180，000英里每秒。在这么一个微小的时间中，粒子的动量，并未被保存。当然，在那一秒之后，电荷感觉到了第一个的影响，所有的事情，尘埃落定后，动量方程将会很好地结账，但是，在那个小的时间间隔内，动量并没有被保存。为了表示这件事情，我们说，在那个时间间隔内，除了粒子的动量即mv之外，还有另外一种形式的动量，也就是说，电磁场中的动量。如果我们把场的动量，加到粒子的动量上，那么，动量就是守恒的，在任一瞬间，一直都守恒。电磁场可以拥有动量和能量这一事实，使得这个电磁场非常真实，于是，为了更好地理解，原始的那个想法--即在粒子之间有力，就要被改成这种想法：一个粒子，形成一个场，而场作用于另一个粒子，这个场本身，就像粒子一样，有一些我们熟悉的属性，如包含能量、和动量。我们看另外一个例子：一个电磁场有波，我们称它为光；结果证实，光也带有动量，于是，当光对一个对象有明显影响时，它每秒都载着一定量的动量，这一点与力一样，因为如果被照明的对象，每秒都在积累一定量的动量，那么，其动量就在改变，这种情况，与一个力作用于其上，几乎一样。通过轰击一个对象，光可以施加压力；这种压力非常小，但是，如果仪器足够精密，还是可以测量的。

Now in quantum mechanics it turns out thatmomentum is a different thing—it is no longer mv . It is hard to define exactly what is meant by the velocity of a particle,but momentum still exists. In quantum mechanics the difference is that when theparticles are represented as particles, the momentum is still mv , but when the particles are represented as waves, the momentum ismeasured by the number of waves per centimeter: the greater this number ofwaves, the greater the momentum. In spite of the differences, the law of conservationof momentum holds also in quantum mechanics. Even though the law F=mais false, and all the derivations of Newton were wrong for theconservation of momentum, in quantum mechanics, nevertheless, in the end, thatparticular law maintains itself!
现在，在量子力学中，结果就是，动量是一个不同的事物，而不再是mv了。一个粒子的矢速，究竟意味着什么，很难定义，但是，动量仍然存在。在量子力学中，区别就是，当粒子是被作为粒子来表现时，动量就仍是mv，但是，当粒子是作为波来表现时，动量就是通过每厘米波的数目，来测量的，这个波的数目越大，动量就越大。除了这点之外，动量守恒的规律，在量子力学中，照样成立。在量子力学中，尽管规律F=ma是错的，对于动量守恒来说，所有牛顿规律的推论都是错的，但是最终，粒子的规律，还能成立！
脚注
1. H. V. Neher and R. B. Leighton, Amer. Jour. of Phys. 31,255 (1963).
1、内赫，和雷顿，《物理学期刊》，31,255（1963）。

1 Chapter11.Vectors第11章 矢量

11–1Symmetry in physics 11-1 物理学中的对称
In this chapter we introduce a subject thatis technically known in physics as symmetry in physical law. The word“symmetry” is used here with a special meaning, and therefore needs to bedefined. When is a thing symmetrical—how can we define it? When we have a picturethat is symmetrical, one side is somehow the same as the other side. Professor HermannWeyl has given this definition of symmetry: a thing is symmetrical if one cansubject it to a certain operation and it appears exactly the same after theoperation. For instance, if we look at a silhouette of a vase that isleft-and-right symmetrical, then turn it 180∘ around the vertical axis, it looks the same. We shall adopt thedefinition of symmetry in Weyl’s more general form, and in that form we shalldiscuss symmetry of physical laws.
在这一章，我们要介绍的主题，是物理规律中的对称，它在物理学中有严格的意义。单词“对称”，在这里有着特殊的意义因此需要定义。一个事物，什么时候才是对称的--我们如何才能定义它呢？如果我们有一张图片，它是对称的，那么，它的一边，在某种意义上，与另一边是一样的。赫尔曼威尔教授，曾给出了这个对称的定义：一个事物，如果可以让它从属于某一操作，在此操作之后，它看上去，几乎一样，那么它就是对称的。例如，我们看一个左右对称的花瓶的轮廓像，然后，把它绕着垂直轴，转180度，它看起来是一样的。我们将在赫尔曼威尔的更广泛的形式上，采取这个关于对称的定义，我们将在这个形式上，讨论物理规律的对称。

Suppose we build a complex machine in a certainplace, with a lot of complicated interactions, and balls bouncing around withforces between them, and so on. Now suppose we build exactly the same kind ofequipment at some other place, matching part by part, with the same dimensionsand the same orientation, everything the same only displaced laterally by somedistance. Then, if we start the two machines in the same initial circumstances,in exact correspondence, we ask: will one machine behave exactly the same asthe other? Will it follow all the motions in exact parallelism? Of course theanswer may well be no, because if we choose the wrong place for ourmachine it might be inside a wall and interferences from the wall would make themachine not work.
假设我们在某地建，建立一种复杂的机器，有很多复杂的交互作用，球在它们之间，来回有力地反弹着，等等。现在，假设我们在另外一个地方，建立同样的设备，每一部分都匹配，有着同样的维度，和同样的方向，所有方面都一样，只是放在前者侧面，有些距离。因此，如果我们在相同的初始条件下，以精确的相应{每种条件都一样？}，启动这两台机器，我们问：一台机器的表现，会与另一台完全一样吗？它会以完全相似的方式，做所有的运动吗？当然，答案完全可能是否，因为，如果我们给我们的机器，选择了错误的地点，它可能在一堵墙里面，来自墙的干扰，可能会让机器无法工作。

All of our ideas in physics require a certainamount of common sense in their application; they are not purely mathematicalor abstract ideas. We have to understand what we mean when we say that thephenomena are the same when we move the apparatus to a new position. We meanthat we move everything that we believe is relevant; if the phenomenon is notthe same, we suggest that something relevant has not been moved, and we proceedto look for it. If we never find it, then we claim that the laws of physics donot have this symmetry. On the other hand, we may find it—we expect to find it—ifthe laws of physics do have this symmetry; looking around, we may discover, forinstance, that the wall is pushing on the apparatus. The basic question is, ifwe define things well enough, if all the essential forces are included insidethe 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?
我们关于物理学的所有想法，在它们的应用中，都要求一定量的常识；它们并不是纯粹数学的或抽象的想法。当我们说：‘当我们把仪器移到一个新的位置时，现象是一样的’，这种说法意味着什么，我们必须要理解。我们的意思是，所有我们相信是有关的东西，我们都移动了；如果现象并不一样，我们会认为，某个相关的东西，并未被移动，我们会去寻找它。如果我们永远都找不到它，那么，我们将会声明，物理学的规律，没有这种对称。另一方面，如果物理学的规律确实有这种对称，那么，我们可能会发现它，这也是我们的期待；例如，环顾四周，我们可能会发现，墙可能正在阻拦着仪器。基本的问题就是，如果我们把事物，定义地足够好，如果所有基本的力，都被包含在了仪器内，如果所有相关部分，都被从一个地方，移动到另一个地方，那么，规律还会一样吗？这个机器装置还会以同样的方式工作吗?

It is clear that what we want to do is tomove all the equipment and essential influences, but not everythingin the world—planets, stars, and all—for if we do that, we have the samephenomenon again for the trivial reason that we are right back where westarted. No, we cannot move everything. But it turns out in practicethat with a certain amount of intelligence about what to move, the machinery willwork. In other words, if we do not go inside a wall, if we know the origin ofthe outside forces, and arrange that those are moved too, then the machinery willwork the same in one location as in another.
有一点很清楚，我们想移动的，是所有的装备和基本影响，而不是世界上的所有事物—行星、恒星、及所有事物—因为，如果我们那样做，我们所拥有的现象，是会一样，原因很简单，我们只是回到了起点而已。不，我们不能移动所有事物。从实践上看，结果就是，对于什么应该移动，使用了一定的智慧后，此机械装置就将工作。换句话说，如果我们不走进墙内，如果我们知道外部力的起源，并且安排好那些被移动的东西{？}，那么，此机械装置，在一个位置上的工作运行，将与在另一位置上的一样。

11–2Translations 11-2 转换
We shall limit our analysis to just mechanics,for which we now have sufficient knowledge. In previous chapters we have seenthat the laws of mechanics can be summarized by a set of three equations foreach particle:
我们的分析，将只局限于力学，对它我们现在有充分的知识。在前面的章节中，我们已经看到，每个粒子的力学规律，可以被总结为一组方程，共三个：
m(d2x/dt2)=Fx, m(d2y/dt2)=Fy, m(d2z/dt2)=Fz. (11.1)
Now this means that there exists a way to measure x , y , and z on three perpendicular axes, and the forces along those directions,such that these laws are true. 现在，这就意味着，存在一种方法，可以测量在三个互相垂直的轴上的x , y ,和z，及沿着这些方向的力，这样，这些规律就是真的。

These must be measured from some origin,but where do we put the origin? All that Newton would tell us at firstis that there is some place that we can measure from, perhaps the centerof the universe, such that these laws are correct. But we can show immediatelythat we can never find the center, because if we use some other origin it wouldmake no difference. In other words, suppose that there are two people—Joe, whohas an origin in one place, and Moe, who has a parallel system whose origin is somewhereelse (Fig. 11–1). Nowwhen Joe measures the location of the point in space, he finds it at x , y , and z (we shall usually leave z out because it is too confusing to draw in a picture). Moe, on theother hand, when measuring the same point, will obtain a different 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
x′=x−a, y′=y, z′=z. (11.2)
这些值，应该从某个原点开始测量，但是，原点放在什么地方呢？牛顿所能告诉我们的，首先就是，有某个地方，我们可以从它开始测量，或许是宇宙的中心，这样，这些规律就是正确的。但是，我们可以立即指出，我们永远也找不到那个中心，因为，如果我们使用某个其他的原点，不会有何不同。换句话说，假设有两个人，一个是Joe，他把一个地方当原点，一个是Moe，他有一个平行的系统，但原点在另一个地方（图11-1）。现在，当Joe测量空间中一个点的位置时，他发现，点在x , y , 和z（通常我们不画z轴，容易引起困惑）。另一方面，当Moe在测量同一个点时，他会得到不同的x (为了区别，我们将称之为 x′)，原则上也可得到一个不同的 y，虽然在我们的例子中，它们在数值上是相等的。于是我们有：
x′=x−a, y′=y, z′=z. (11.2)