The reason we bring this out is that theidea of force is not particularly suitable for quantum mechanics; therethe idea of energy is most natural. We find that although forces andvelocities “dissolve” and disappear when we consider the more advanced forcesbetween nuclear matter and between molecules and so on, the energy conceptremains. Therefore we find curves of potential energy in quantum mechanicsbooks, but very rarely do we ever see a curve for the force between twomolecules, because by that time people who are doing analyses are thinking interms of energy rather than of force.
我们为什么要提出这点，乃是因为，力的想法，并不特别适合于量子力学，在量子力学中，能量的想法，最自然。我们发现，当我们考虑原子核之间及分子之间等的更先进的力时，虽然力和矢速的概念“分解了”并消失了，但能量的概念还保留着。因此，我们在量子力学的书中，发现了势能的曲线，但是，对于两个分子之间，我们很少看到关于力的曲线，因为，在这个地方，做分析的人，与其说是用力在考虑，不如说是用能量在考虑。

Next we note that if several conservativeforces are acting on an object at the same time, then the potential energy ofthe object is the sum of the potential energies from each of the separateforces. This is the same proposition that we mentioned before, because if theforce can be represented as a vector sum of forces, then the work done by thetotal force is the sum of the works done by the partial forces, and it cantherefore be analyzed as changes in the potential energies of each of themseparately. Thus the total potential energy is the sum of all the little pieces.
下面我们注意到，如果几个保守力，同时作用于一个对象，那么，这个对象的势能，就是每一个单独的力的势能的总和。这与我们以前提到的命题一样，因为，如果力可以被表示为几个力的矢量和，那么，总的力所做的功，就是每一个分力所做的功的总和，因此，它也就可以被分析为每个单独的力的势能。这样，总的势能，就是所有小块的总和。

We could generalize this to the case of asystem of many objects interacting with one another, like Jupiter, Saturn,Uranus, etc., or oxygen, nitrogen, carbon, etc., which are acting with respectto one another in pairs due to forces all of which are conservative. In thesecircumstances the kinetic energy in the entire system is simply the sum of thekinetic energies of all of the particular atoms or planets or whatever, and thepotential energy of the system is the sum, over the pairs of particles, of thepotential energy of mutual interaction of a single pair, as though the otherswere not there. (This is really not true for molecular forces, and the formulais somewhat more complicated; it certainly is true for Newtonian gravitation, andit is true as an approximation for molecular forces. For molecular forces thereis a potential energy, but it is sometimes a more complicated function of thepositions of the atoms than simply a sum of terms from pairs.) In the specialcase of gravity, therefore, the potential energy is the sum, over all the pairsi and j , of −Gmimj/rij , as was indicated in Eq. (13.14). Equation (13.14) expressed mathematically the followingproposition: that the total kinetic energy plus the total potential energy doesnot change with time. As the various planets wheel about, and turn and twist andso on, if we calculate the total kinetic energy and the total potential energywe find that the total remains constant.
我们可以把这一点，推广到：有很多对象相互作用的系统中，例如木星、土星、天王星等，或者氧、氮、碳等，在这些系统中，就一个对象而言，它与其他对象之间，有相互的力，所有这些力，都是保守的。在这些情形中，整个系统的动能，简单地说，就是所有具体原子或行星（或不论什么）的动能的总和，而系统的势能，则是所有‘相互作用的粒子对的势能’的总和，就好像，其他粒子是不存在的一样。（对于分子力，这并非真实情况，相关公式，要更复杂些；对于牛顿万有引力来说，它当然为真；对于分子力，它可以作为一种近似。对于分子力，有势能，但是，它是某种更复杂的原子位置的函数，而不是一个简单的每对粒子的{势能}的总和。）因此，在重力这种特殊情况下，势能就是所有粒子对i和j的−Gmimj/rij的总和，如方程（13.14）所示。方程（13.14）数学地表达了下面的命题：总的动能加上总的势能，不随时间而变。随着不同的行星转过，旋转并扭动，如果我们计算总的动能，和总的势能，就会发现，总量恒为常数。

14–4 Nonconservative forces 14-4 非保守力
We have spent a considerable timediscussing conservative forces; what about nonconservative forces? We shalltake a deeper view of this than is usual, and state that there are nononconservative forces! As a matter of fact, all the fundamental forces innature appear to be conservative. This is not a consequence of Newton’s laws.In fact, so far as Newton himself knew, the forces could be nonconservative, asfriction apparently is. When we say friction apparently is, we aretaking a modern view, in which it has been discovered that all the deep forces,the forces between the particles at the most fundamental level, areconservative.
我们讨论保守力，所花时间，相当可观，那么，非保守力如何呢？我们将对它，做一个比通常更深入的视察，然后声明：没有非保守力！作为一种事实情况，自然中所有基础力的表现，都是保守性的。这不是牛顿规律的后果。事实上，就牛顿本人所知，力可以是非保守的，如摩擦力，显然就是。当我们说摩擦力，明显就是的时候，我们是用了一个现代的视角，在其中，已经发现，所有深的力，如在最基础层面的粒子之间的力，是保守的。

If, for example, we analyze a system likethat great globular star cluster that we saw a picture of, with the thousands ofstars all interacting, then the formula for the total potential energy is simplyone term plus another term, etc., summed over all pairs of stars, and the kineticenergy is the sum of the kinetic energies of all the individual stars. But theglobular cluster as a whole is drifting in space too, and, if we were farenough away from it and did not see the details, could be thought of as asingle object. Then if forces were applied to it, some of those forces mightend up driving it forward as a whole, and we would see the center of the wholething moving. On the other hand, some of the forces can be, so to speak, “wasted”in increasing the kinetic or potential energy of the “particles” inside. Let ussuppose, for instance, that the action of these forces expands the wholecluster and makes the particles move faster. The total energy of the wholething is really conserved, but seen from the outside with our crude eyes whichcannot see the confusion of motions inside, and just thinking of the kineticenergy of the motion of the whole object as though it were a single particle,it would appear that energy is not conserved, but this is due to a lack ofappreciation of what it is that we see. And that, it turns out, is the case:the total energy of the world, kinetic plus potential, is a constant when welook closely enough.
例如，如果我们分析一个系统，比如巨大的球形星簇--我们通过照片看到的那种，它由上千个交互作用的恒星组成，那么，其总势能的公式，简单说就是：一个恒星对的势能，加上另外一个的，如此等等，遍历所有的恒星对，而动能，就是所有个别恒星动能的总和。但是，球形星簇，也是作为一个整体，漂在空中，如果我们离它足够远，看不清细节，那么，它就可以被思考为一个单独的对象。因此，如果有力，作用于它，那么，这些力的一部分，可能会用来驱动它，作为一个整体，向前移动，那么，我们就可以看到，整个星簇的中心，在移动。另一方面，力的另外一部分，这么说吧，可能被“浪费了”，因为它们增加了内部某些粒子的动能和势能。例如，让我们假设，这些力的作用，可以扩展到整个星簇，让粒子移动地更快。整个事物的总能量，确实是保守的，但是，我们的裸眼，看不到内部混乱运动，用我们的裸眼，从外面看，只能把整个对象，当作一个单独的粒子，从而会想，其运动的动能，似乎是不保守的，但是，这要归于，我们并不了解我们所看到的东西。结果就是：对于这个世界，当我们足够近地观察它时，其总能量，即动能加势能，是一个常数。

When we study matter in the finest detailat the atomic level, it is not always easy to separate the total energyof a thing into two parts, kinetic energy and potential energy, and such separationis not always necessary. It is almost always possible to do it, so letus say that it is always possible, and that the potential-plus-kinetic energyof the world is constant. Thus the total potential-plus-kinetic energy insidethe whole world is constant, and if the “world” is a piece of isolated material,the energy is constant if there are no external forces. But as we have seen,some of the kinetic and potential energy of a thing may be internal, forinstance the internal molecular motions, in the sense that we do not notice it.We know that in a glass of water everything is jiggling around, all the parts aremoving all the time, so there is a certain kinetic energy inside, which weordinarily may not pay any attention to. We do not notice the motion of theatoms, which produces heat, and so we do not call it kinetic energy, but heatis primarily kinetic energy. Internal potential energy may also be in the form,for instance, of chemical energy: when we burn gasoline energy is liberatedbecause the potential energies of the atoms in the new atomic arrangement arelower than in the old arrangement. It is not strictly possible to treat heat asbeing pure kinetic energy, for a little of the potential gets in, and viceversa for chemical energy, so we put the two together and say that the totalkinetic and potential energy inside an object is partly heat, partly chemicalenergy, and so on. Anyway, all these different forms of internal energy aresometimes considered as “lost” energy in the sense described above; this willbe made clearer when we study thermodynamics.
对于物质，当我们在原子层这一最详细的部分，来研究它时，把总的能量分成两部分，即动能和势能，并不总是很容易，且这种分开，并非总是必要的。做这种区分，几乎总是可能的，所以，让我们说，这总是可能的，且世界的势能加动能，是一个常数。这样，在一个整体世界的内部，总的势能加动能，是一个常数，如果此“世界”，是一个孤立体，那么，如果没有外部力的话，能量就是常数。但是，正如我们看到的，一个事物的某些动能和势能，可能是内部的，例如内部分子的运动，这种意义，我们并没有注意到。我们知道，在一杯水中，所有东西都在摇动，所有的部分，都一直在运动，所以，其内部就有一定的动能，对此，我们通常并未注意。我们并未注意到原子的运动，它会产生热，所以，我们并没有把它称为动能，但热主要是动能。再例如，内部势能，也可以化学能的形式存在，当我们燃烧汽油时，能量就被释放，因为，原子的势能，在新原子中的排列，比在旧原子中的排列，要低。把热当做纯粹的动能来对待，并不是严格意义上可能的，因为一些势能会进来，对化学能也如此，所以，我们把这两个放在一起，说一个对象中的总的动能和势能，部分是热能，部分是化学能等。总之，所有这些内部能量的不同形式，有时，按照上面所描述的意义，而被考虑为“丢失了”的能量；这一点，当我们研究热力学时，就会更清楚了。

As another example, when friction is presentit is not true that kinetic energy is lost, even though a sliding object stopsand the kinetic energy seems to be lost. The kinetic energy is not lostbecause, of course, the atoms inside are jiggling with a greater amount ofkinetic energy than before, and although we cannot see that, we can measure itby determining the temperature. Of course if we disregard the heat energy, thenthe conservation of energy theorem will appear to be false.
作为另一个例子，当摩擦力在场时，动能消失了这种说法，并不正确，尽管一个滑动对象停止了，且动能似乎消失了。动能没有消失，当然是因为，内部原子，正在用比以前更大的动能，在摇动，虽然我们看不到，但我们可以通过测温，来计量它。当然，如果我们忽视热能，那么，能量守恒原理，似乎就是错的。

Another situation in which energy conservationappears to be false is when we study only part of a system. Naturally, theconservation of energy theorem will appear not to be true if something is interactingwith something else on the outside and we neglect to take that interaction intoaccount.
还有一种情况，在其中，能量守恒看上去是错的，这是因为我们研究的，只是系统的一部分。如果外部有某物与某物交互作用，而我们却没有把这种交互计算在内，那么自然，能量守恒原理，看上去就不正确。

14–5Potentials and fields 14-5 势与场
We shall now discuss a few of the ideas associatedwith potential energy and with the idea of a field. Suppose we have twolarge objects A and B and a third very small one which is attracted gravitationally by thetwo, with some resultant force F . We have already noted in Chapter 12that the gravitational force on a particle can be written as its mass, m, times another vector, C , which is dependent only upon the position of the particle:
F=mC.
We can analyze gravitation, then, by imagining that there is a certainvector C at every position in space which “acts” upon a mass which we may placethere, but which is there itself whether we actually supply a mass for it to“act” on or not. C has three components, and each of those components is a functionof (x,y,z) , a function of position in space. Such a thing we call a field,and we say that the objects A and B generate the field, i.e., they “make” the vector C. When an object is put in a field, the force on it is equal to itsmass times the value of the field vector at the point where the object is put.
现在，我们将讨论几个与势能有关的想法，及与场有关的想法。假设我们有两个大的对象A和B，及第三个非常小的对象，它被前两个的万有引力的某种合力F所吸引。在第12章，我们已经注释了，作用于一个粒子上的万有引力，可被写为：其质量m，乘以另一个矢量C，它只依赖于粒子的位置：
F=mC.
因此，我们可以这样分析万有引力，即通过想象，在空间中的每个位置，都有某个矢量C，它会“作用于”我们放在那里的质量，但是，无论我们是否在那里放一个质量，它本身总是在那里的。C有三个分量，每个分量，都是(x,y,z)的函数，即是一个空间位置的函数。这样一种事物，我们称为场，我们说，对象A和B产生了场，亦即，它们造成了矢量C。当一个对象被放入场中时，作用于其上的力，就等于，其质量，乘以它所在的位置的场的矢量。

We can also do the same with the potentialenergy. Since the potential energy, the integral of (−force)⋅(ds) can be written as m times the integral of (−field)⋅(ds) , a mere change of scale, we see that the potential energy U(x,y,z)of an object located at a point (x,y,z) in space can be written as m times another function which we may call the potential Ψ . The integral ∫C⋅ds=−Ψ , just as ∫F⋅ds=−U ; there is only a scale factor between the two:
U=−∫F⋅ds=−m∫C⋅ds=mΨ. (14.7)
对于势能，我们也可以做同样的事。由于势能，即 (−force)⋅(ds)的积分，也可被写成：m乘以 (−force)⋅(ds)的积分，这仅仅一个标量的变化，我们看到，一个位于空间中点 (x,y,z)处的对象的势能U(x,y,z)，可以被写成m乘以另外一个函数，这我们称为势函数Ψ。积分 ∫C⋅ds=−Ψ ，正如积分 ∫F⋅ds=−U一样；两者之间，只差一个标量因子：
U=−∫F⋅ds=−m∫C⋅ds=mΨ. (14.7)

By having this function Ψ(x,y,z) at every point in space, we can immediately calculate the potentialenergy of an object at any point in space, namely, U(x,y,z)=mΨ(x,y,z)—rather a trivial business, it seems. But it is not really trivial,because it is sometimes much nicer to describe the field by giving the valueof Ψ everywhere in space instead of having to give C . Instead of having to write three complicated components of a vectorfunction, we can give instead the scalar function Ψ . Furthermore, it is much easier to calculate Ψ than any given component of C when the field is produced by a number of masses, for since thepotential is a scalar we merely add, without worrying about direction. Also,the field C can be recovered easily from Ψ , as we shall shortly see. Suppose we have point masses m1 , m2 , … at the points 1 , 2 , … and we wish to know the potential Ψ at some arbitrary point p . This is simply the sum of the potentials at p due to the individual masses taken one by one:
通过让空间中的每一点，都有函数Ψ(x,y,z)，那么，对于空间中任意一点上的对象，我们都可以立即计算其势能，即U(x,y,z)=mΨ(x,y,z)，这件事情，似乎有点微不足道。但是，它并非微不足道，因为，有时候，对于空间中的每一点，给出其Ψ的值，而不是给出其 C值，要好得多。我们只需给出标量函数Ψ，而不用写出有三个分量的、复杂的矢量函数。另外，当场是由若干质量所产生的时候，计算Ψ，比计算任何被给予的C的分量，要容易得多，因为势是一个标量，我们只需要加就行了，而不用担心方向。另外，根据Ψ来恢复场C，也很容易，这我们很快就会看到。假设我们在点1 , 2 , …，有质量m1，m2 , …，对于任意点p，我们希望知道其势Ψ。这就是每一个质量在点p所产生的势的总和：
(14.8)

Fig. 14–4.Potential due to a sphericalshell of radius a . 图 14-4 可归于半径为a的球形壳的势。
In the last chapter we used this formula,that the potential is the sum of the potentials from all the different objects,to calculate the potential due to a spherical shell of matter by adding thecontributions to the potential at a point from all parts of the shell. Theresult of this calculation is shown graphically in Fig. 14–4. Itis negative, having the value zero at r=∞ and varying as 1/r down to the radius a , and then is constant inside the shell. Outside the shell thepotential is −Gm/r , where m is the mass of the shell, which is exactly the same as it would havebeen if all the mass were located at the center. But it is not everywhereexactly the same, for inside the shell the potential turns out to be −Gm/a, and is a constant! When the potential is constant, there is nofield, or when the potential energy is constant there is no force, becauseif we move an object from one place to another anywhere inside the sphere thework done by the force is exactly zero. Why? Because the work done in movingthe object from one place to the other is equal to minus the change in thepotential energy (or, the corresponding field integral is the change of thepotential). But the potential energy is the same at any two pointsinside, so there is zero change in potential energy, and therefore no work isdone in going between any two points inside the shell. The only way the workcan be zero for all directions of displacement is that there is no force atall.
再上一章，我们用过这个公式，势就是所有不同对象所产生的势的总和，要计算一个球形壳所产生的势，就是把壳上所有部分，对一个点的势的贡献，全加起来。这个计算的结果，如图14-4所示。在 r=∞处，它是负的，随着1/r变化，直到半径为a处，然后，在壳的内部，是常数。在壳的外面，势是−Gm/r ，这里m是壳的质量，这个势，与球壳质量全集中在中心时，是一样的。但是，并不是处处都完全一样，因为在壳的内部，势就变成了−Gm/a，是一个常数！当势是一个常数时，就没有场，或者，当势能是一个常数时，就没有力，因为，如果在球内，我们把一个对象，从一个地方挪到另外一个地方，力所做的功就是零。为什么？因为把一个对象，从一个地方挪到另外一个地方，所做的功，等于负的势能的变化（或者相关场的积分，就是势的变化）。但是，在内部，任意两点的势能都一样，所以，势能的变化就是零，因此，在球壳内，从一点到另一点，并不做功。对于任意方向的位移，功为零的唯一方法，就是根本没有力。

This gives us a clue as to how we canobtain the force or the field, given the potential energy. Let us suppose thatthe potential energy of an object is known at the position (x,y,z)and we want to know what the force on the object is. It will not do toknow the potential at only this one point, as we shall see; it requiresknowledge of the potential at neighboring points as well. Why? How can wecalculate the x -component of the force? (If we can do this, of course, we can alsofind the y - and z -components, and we will then know the whole force.) Now, if we wereto move the object a small distance Δx , the work done by the force on the object would be the x -component of the force times Δx , if Δx is sufficiently small, and this should equal the change in potentialenergy in going from one point to the other:
ΔW=−ΔU=FxΔx. (14.9)
We have merely used the formula ∫F⋅ds=−ΔU , but for a very short path. Now we divide by Δx and so find that the force is
Fx=−ΔU/Δx. (14.10)
这给了我们一条线索，即势能被给予了，如何得到力或场。我们假设，一个对象，在点(x,y,z)，其势能已知，我们想知道：作用于其上的力是什么。我们将会看到，只知道这一个点的势，是不行的；对于周围的点，也需要其势的知识。为什么？我们如何计算力的x分量呢？（如果我们可以做到这点，当然，我们就可以同样得到y方向和z方向的分量，然后，我们就可以知道整个力。）现在，如果我们把对象，移动一个小的距离Δx，那么，对于对象所做的功，将是x方向的力乘以Δx，如果Δx足够小，那么，这就应该等于：从一个点到另一个点的势能的变化：
ΔW=−ΔU=FxΔx. (14.9)
我们只是用了公式∫F⋅ds=−ΔU，但只是对一个非常短的距离。现在，除以Δx，就得到力：
Fx=−ΔU/Δx. (14.10)

Of course this is not exact. What we really want is the limitof (14.10)as Δx gets smaller and smaller, because it is only exactly right inthe limit of infinitesimal Δx . This we recognize as the derivative of U with respect to x , and we would be inclined, therefore, to write −dU/dx. But U depends on x , y , and z , and the mathematicians have invented a different symbol to remind usto be very careful when we are differentiating such a function, so as toremember that we are considering that only x varies, and y and z do not vary. Instead of a d they simply make a “backwards 6 ,” or ∂ . (A ∂ should have been used in the beginning of calculus because we alwayswant to cancel that d , but we never want to cancel a ∂ !) So they write ∂U/∂x , and furthermore, in moments of duress, if they want to be verycareful, they put a line beside it with a little yz at the bottom (∂U/∂x|yz ), which means “Take the derivative of U with respect to x , keeping y and z constant.” Most often we leave out the remark about what is keptconstant because it is usually evident from the context, so we usually do notuse the line with the y and z . However, always use a ∂ instead of a d as a warning that it is a derivative with some other variables keptconstant. This is called a partial derivative; it is a derivative inwhich we vary only x .
当然，这并不准确。我们真正想要的，是当Δx变得越来越小时，（14.10）的极限，因为，它只在间隔Δx取极限时，才完全正确。我们把这个，认为是U对x的导数，因此，我们倾向于写−dU/dx。但是, U 依赖于 x , y , 和z，所以，数学家们发明了一个不同的符号，提醒我们，当我们对这种函数求微分时，要非常小心，为的是记住，我们考虑的只是x的变化，而y和z，并没有变。为了代替d，他们简单地发明了一个“向后的6”，或者∂.（在微积分开始时，就应该用到∂，因为，我们总是想消去那个d，但是，我们永远也不会想着消去∂！{？}）于是，它们写出∂U/∂x,更进一步，在需要强迫时，如果他们想非常小心，他们会在底部，放一条线，和小的yz，(∂U/∂x|yz)，其意思是“求U对x 的导数，y和 z保持为常数。”大多数情况下，那些需要被保持为常数的项，我们不会写，因为，通常从上下文看，很明显，所以，我们通常不用线及y和 z。然而，总是用 ∂ 代替 d，作为一种警告：这个导数中，有些变量保持为常数。这被称为偏导数；在这个导数中，我们只变更x。

Therefore, we find that the force in the x-direction is minus the partial derivative of U with respect to x :
Fx=−∂U/∂x. (14.11)
In a similar way, the force in the y -direction can be found by differentiating U with respect to y , keeping x and z constant, and the third component, of course, is the derivative withrespect to z , keeping y and x constant:
Fy=−∂U/∂y, Fz=−∂U/∂z. (14.12)
This is the way to get from the potential energy to the force. We getthe field from the potential in exactly the same way:
Cx=−∂Ψ/∂x, Cy=−∂Ψ/∂y, Cz=−∂Ψ/∂z. (14.13)
因此，我们发现，x方向的力，就是负的 U对x的偏导数:
Fx=−∂U/∂x. (14.11)
类似地，y方向的力，就是负的 U对y的偏导数，x和z保持为常数，而第三个分量，当然就是对z的偏导数，y和x保持为常数：
Fy=−∂U/∂y, Fz=−∂U/∂z. (14.12)
这就是从势能出发，得到力的方法。我们从势出发，以完全同样的方式，得到场：
Cx=−∂Ψ/∂x, Cy=−∂Ψ/∂y, Cz=−∂Ψ/∂z. (14.13)