To conclude this discussion, let usdescribe qualitatively what happens if we proceed further in analyzing a linearproblem with a given force, when the force is quite complicated. Out of themany possible procedures, there are two especially useful general ways that wecan solve the problem. 要总结这个讨论，让我们定性地描述一下，在下面的情况中，会发生什么，这种情况就是：设有一个线性问题，带有一个被给予的力，当此力相当复杂时，我们还要进一步分析此问题。要解这个问题，有很多可能的过程，其中有两种普遍方法，特别有用。One is this: suppose that we can solve it for special known forces,such as sine waves of different frequencies. We know it is child’s play tosolve it for sine waves. So we have the so-called “child’s play” cases. Now thequestion is whether our very complicated force can be represented as the sum oftwo or more “child’s play” forces. 一种是：假设对于具体已知的力，例如不同频率的正弦波，我们可以解它。我们知道，解正弦波这种问题，属于小孩把戏。所以，我们就有了所谓的“小孩把戏”的情况。现在的问题就是，我们非常复杂的力，是否可被表现为：两个以上的“小孩把戏”的力的总和。In Fig. 25–1 wealready had a fairly complicated curve, and of course we can make it morecomplicated still if we add in more sine waves. So it is certainly possible toobtain very complicated curves. 在图25-1中，我们已经有了一个相当复杂的曲线，当然，如果我们给它，加入更多正弦波的话，那么，我们也落让它，变得更复杂。所以，要得到更复杂的曲线，当然是可能的。And, in fact, the reverse is also true: practically every curve can beobtained by adding together infinite numbers of sine waves of differentwavelengths (or frequencies) for each one of which we know the answer. 事实上，相反的过程，也是真的：实践上，每条曲线，都可通过，把无限数目的不同波长（或频率）的正弦波，加在一起，而得到；对于每个正弦波，我们都知道它的解答。We just have to know how much of each sine wave to put in to makethe given F , and then our answer, x , is the corresponding sum of the F sine waves, each multiplied by its effective ratio of x to F .我们只需要知道，要生成被给予的力F，每个正弦波，要加入多少，因此，我们的答案x，就是相应的F正弦波的总和{？}，每个都要乘以它的x对F的有效比率。 This method of solution is called the method of Fouriertransforms or Fourier analysis. We are not going to actually carryout such an analysis just now; we only wish to describe the idea involved.
这种解的方法，被称为傅立叶变换、或傅立叶分析。现在，我们还不会拿出这样一个分析；我们只是希望，描述所牵扯到的想法。

Another way in which our complicatedproblem can be solved is the following very interesting one. Suppose that, bysome tremendous mental effort, it were possible to solve our problem for aspecial force, namely an impulse. The force is quickly turned on andthen off; it is all over. Actually we need only solve for an impulse of some unitstrength, any other strength can be gotten by multiplication by an appropriatefactor. We know that the response x for an impulse is a damped oscillation. Now what can we say about someother force, for instance a force like that of Fig. 25–4?
我们的复杂问题，还有一种解法，非常有趣，就是下面这种方法。假设通过巨大的心智努力，为一种特殊的力，比如脉冲，来解我们的问题，是可能的。力会被很快地加上，然后又去掉；然后就结束了。实际上，我们要解的，只是一些单位强度的脉冲，其他强度的，可以通过乘以一个合适的因子，来得到。我们知道，对脉冲的反应x，是一个阻尼振荡。现在，关于某些其他的力，例如，图25-4所示的力，我们能说些什么呢？

Fig. 25–4.A complicated force may betreated as a succession of sharp impulses. 图25-4 一个复杂的力，可能被当作尖锐脉冲的相继，来对待。

Such a force can be likened to a successionof blows with a hammer. First there is no force, and all of a sudden there is asteady force—impulse, impulse, impulse, impulse, … and then it stops. Inother words, we imagine the continuous force to be a series of impulses, veryclose together. Now, we know the result for an impulse, so the result for awhole series of impulses will be a whole series of damped oscillations: it willbe the curve for the first impulse, and then (slightly later) we add to thatthe curve for the second impulse, and the curve for the third impulse, and soon. Thus we can represent, mathematically, the complete solution for arbitraryfunctions if we know the answer for an impulse. We get the answer for any otherforce simply by integrating. This method is called the Green’s functionmethod. A Green’s function is a response to an impulse, and the method ofanalyzing any force by putting together the response of impulses is called theGreen’s function method.
这种力，可以比作，用一个锤子，连续击打。开始没有力，突然就有了一个稳定的力--脉冲、脉冲、脉冲、脉冲…，然后，又停止了。换句话说，我们可以把连续的力，想象为是一个脉冲的序列，脉冲之间，非常接近。现在，我们知道了一个脉冲的结果，所以，全部脉冲系列的结果，就是全部阻尼振荡的系列：首先，是第一个脉冲的曲线，然后（稍微晚点），我们给这个曲线，加上第二个脉冲的，然后，是第三个脉冲的，如此等等。这样，如果我们知道了一个脉冲的答案，那么，对于任意的函数，我们都可以表示其数学上的完整解。对于任何其他的力，我们只要通过集成，就可以得到其答案。这个方法，被称为格林函数法。格林函数，是对一个脉冲的反应；对于任何力，通过把诸脉冲的反应，放在一起，来分析它的方法，就被称为格林函数法。

The physical principles involved in both ofthese schemes are so simple, involving just the linear equation, that they canbe readily understood, but the mathematical problems that are involved,the complicated integrations and so on, are a little too advanced for us toattack right now. You will most likely return to this some day when you havehad more practice in mathematics. But the idea is very simple indeed.
在这两个方案中，所牵扯到的物理原理，非常简单，只是线性方程，很易理解，也可欣然接受，但是，所牵扯到的数学方面的问题，如复杂的积分问题等，让我们现在来解决的话，有点儿超前。未来某天，当你们在数学方面，有了更多的实践时，你可能会喜欢，回到这里。但是，这里所牵扯到的想法，确实非常简单。

Finally, we make some remarks on why linearsystems are so important. The answer is simple: because we can solve them! Somost of the time we solve linear problems. Second (and most important), it turnsout that the fundamental laws of physics are often linear. The Maxwellequations for the laws of electricity are linear, for example. The great lawsof quantum mechanics turn out, so far as we know, to be linear equations. Thatis why we spend so much time on linear equations: because if we understandlinear equations, we are ready, in principle, to understand a lot of things.
最后，对于线性系统，为什么如此重要，我们要做一些说明。答案很简单，因为我们能解它们。所以，大部分时间，我们可以解线性问题。第二（也是最重要的），结果显示，物理学的基础规律，通常是线性的。例如，麦克斯韦方程组，是关于电规律的，它是线性的。伟大的量子力学的规律，就我们所知，也是线性方程组。这就是为什么，我们要在花大量的时间，在线性方程上：因为，如果我们理解了线性方程，那么，我们就已经可以从原理上，理解很多事情了。

We mention another situation where linearequations are found. When displacements are small, many functions can be approximatedlinearly. For example, if we have a simple pendulum, the correct equation forits motion is
我们提到过另外一种情况，在那里，也发现了线性方程。当位移很小时，很多函数，是近似线性的。例如，如果我们有一个简单的单摆，那么，其运动的正确方程就是：
d2θ/dt2=−(g/L)sinθ. (25.9)
This equation can be solved by elliptic functions, but the easiest wayto solve it is numerically, as was shown in Chapter 9on Newton’s Laws of Motion. A nonlinear equation cannot be solved, ordinarily,any other way but numerically. Now for small θ , sinθ is practically equal to θ , and we have a linear equation. It turns out that there are manycircumstances where small effects are linear: for the example here the swing ofa pendulum through small arcs. As another example, if we pull a little bit on aspring, the force is proportional to the extension. If we pull hard, we breakthe spring, and the force is a completely different function of the distance!Linear equations are important. In fact they are so important that perhapsfifty percent of the time we are solving linear equations in physics and inengineering.
这个方程，可以通过椭圆函数来解，但是，解它的最容易的方法，是数值式地，在关于牛顿运动定律的第九章，我们曾指出过。通常，一个非线性的方程，只有数值解，而没有其他的解。现在，对于小的θ , sinθ实际上就等于θ，这样，我们就有了一个线性方程。结果显示，在很多情形中，小的作用，是线性的：比如这个例子中，单摆的摆绳，通过小的圆弧。另一个例子，如果我们把弹簧，稍微拉一点，那么，这个力，就正比于拉伸。如果我们用力拉，把弹簧拉断了，那么，力就是位移的一个完全不同的函数。线性方程，非常重要。事实上，它们是如此重要，以至于，在物理学和工程中，我们50%的时间，是在解线性方程。

25–3Oscillations in linear systems 25-3 线性系统中的震荡
Let us now review the things we have beentalking about in the past few chapters. It is very easy for the physics ofoscillators to become obscured by the mathematics. The physics is actually verysimple, and if we may forget the mathematics for a moment we shall see that wecan understand almost everything that happens in an oscillating system. First,if we have only the spring and the weight, it is easy to understand why the systemoscillates—it is a consequence of inertia. We pull the mass down and the forcepulls it back up; as it passes zero, which is the place it likes to be, itcannot just suddenly stop; because of its momentum it keeps on going and swingsto the other side, and back and forth. So, if there were no friction, we wouldsurely expect an oscillatory motion, and indeed we get one. But if thereis even a little bit of friction, then on the return cycle, the swing will notbe quite as high as it was the first time.
现在，让我们回顾一下，我们前面几章讨论过的事情。通过数学，让震荡中的物理，变得模糊，非常容易。物理实际上非常简单，对于一个震荡系统中所发生的事情，如果我们把数学，暂时放下，那么，我们就可以看到，我们可以理解所有这些事情。首先，如果我们只有弹簧和重量，那么，就很容易理解，为什么系统会振荡--它是惯性的后果。我们把质量往下拉，然后，力把它往回拉，在它通过零点、也就是它通常所在的位置时，它不可能突然停下来，由于惯性，它将继续走，并且，摆回到另外一侧，如此往返。所以，如果没有摩擦，那么，我们肯定会期待一个震荡运动，确实，我们也得到了一个。但是，如果哪怕只有一点点摩擦，那么，在回转周期上，摆动就将不会像第一次那么高。

What happens if the friction is not so artificial;for example, ordinary rubbing on a table, so that the friction force is acertain constant amount, and is independent of the size of the oscillation thatreverses its direction each half-cycle? Then the equation is no longer linear,it becomes hard to solve, and must be solved by the numerical method given inChapter 9, or by considering each half-cycle separately. Thenumerical method is the most powerful method of all, and can solve anyequation. It is only when we have a simple problem that we can use mathematicalanalysis.
如果摩擦力的人造性，不是这么强，那么，会发生什么呢？例如，桌面上的普通摩擦，这种摩擦力，就是某个常数量，独立于振幅的大小，且每半个循环，就改变的方向{？}。因此，方程就不再是线性的，变得难解，必须通过在第九章所给数值方法来解，或者，通过分开考虑每半个循环来解。数值方法，是所有方法中最强有力的方法，可以解任何方程。只有在问题简单时，我们才能用数学分析的方法。

Mathematical analysis is not the grandthing it is said to be; it solves only the simplest possible equations. As soonas the equations get a little more complicated, just a shade—they cannot besolved analytically. But the numerical method, which was advertised at thebeginning of the course, can take care of any equation of physical interest.
数学分析，并不像所说的那么壮丽堂皇，它只能解最简单的可能的方程。只要方程变得稍微复杂一点，只需有点阴影—都不能分析性地去解。但是，在本课程开始时，我们展示过的数值方法，则可以解任何物理感兴趣的方程。

Next, what about the resonance curve? Whyis there a resonance? First, imagine for a moment that there is no friction,and we have something which could oscillate by itself. If we tapped thependulum just right each time it went by, of course we could make it go likemad. But if we close our eyes and do not watch it, and tap at arbitrary equalintervals, what is going to happen? Sometimes we will find ourselves tappingwhen it is going the wrong way. When we happen to have the timing just right,of course, each tap is given at just the right time, and so it goes higher andhigher and higher. So without friction we get a curve which looks like thesolid curve in Fig. 25–5 fordifferent frequencies. Qualitatively, we understand the resonance curve; inorder to get the exact shape of the curve it is probably just as well to do themathematics. The curve goes toward infinity as ω→ω0, where ω0 is the natural frequency of the oscillator.
下面来看，共振曲线又如何呢？为什么会有共振？首先，想象一下，若没有摩擦，那么，我们有个东西，它可以自己震荡。对于单摆，如果它每次摆过来时，我们都在正确的时间，拍它，那么当然，我们就可以让它走的很疯狂。但是，如果我们闭上眼睛不看，然后，在任意相等的时间间隔，随机地拍它，那么，会发生什么呢？有时，我们会发现，我们拍在了错误的方向。当我们的计时，碰巧正确时，当然，每次都在正确的时间拍，所以，单摆就会越来越高，越来越高。所以，如果没有摩擦，那么，对于不同的频率，我们得到的曲线，看上去，就像图25-5中的实线一样。定性地说，我们理解共振曲线；为了得到曲线的准确形状，用数学求解也同样很好。当 ω→ω0时，曲线就趋向于无穷大， ω0是震荡的自然频率。

Fig. 25–5.Resonance curves with variousamounts of friction present. 图25-5 现场，使用不同的摩擦量，所提供的共振曲线。

Finally, we take the case where there is anenormous amount of friction. It turns out that if there is too much friction,the system does not oscillate at all. The energy in the spring is barely ableto move it against the frictional force, and so it slowly oozes down to theequilibrium point.
最后，让我们看看，摩擦力巨大这种情况。结果显示，如果摩擦力太大，那么，系统根本就不会震荡。弹簧中的能量，几乎无**服摩擦力，来移动弹簧，所以，弹簧就会慢慢地移向平衡点。

25–4Analogs in physics 25-4 物理中的类比
The next aspect of this review is to notethat masses and springs are not the only linear systems; there are others. Inparticular, there are electrical systems called linear circuits, in which wefind a complete analog to mechanical systems. We did not learn exactly whyeach of the objects in an electrical circuit works in the way it does—that isnot to be understood at the present moment; we may assert it as anexperimentally verifiable fact that they behave as stated.
这个回顾的下一方面，就是要说明，质量和弹簧，并不是唯一的线性系统；还有其他的。特别是，有些电子系统，被称为线性电路，在其中，我们发现了，完全可与机械系统类比的东西。为什么电路中的每个对象，都以这种方式工作，我们并没有学习—当前，这还难以理解；它们的表现，确实如我们所说{是线性的}，我们可以把这，作为一种可实验验证的事实，来断言。

For example, let us take the simplestpossible circumstance. We have a piece of wire, which is just a resistance, andwe have applied to it a difference in potential, V . Now the V means this: if we carry a charge q through the wire from one terminal to another terminal, the work doneis qV . The higher the voltage difference, the more work was done when thecharge, as we say, “falls” from the high potential end of the terminal to thelow potential end. So charges release energy in going from one end to the other.Now the charges do not simply fly from one end straight to the other end; theatoms in the wire offer some resistance to the current, and this resistanceobeys the following law for almost all ordinary substances: if there is acurrent I , that is, so and so many charges per second tumbling down, the numberper second that comes tumbling through the wire is proportional to how hard wepush them—in other words, proportional to how much voltage there is:
例如，让我们取最简单的可能的情形。我们有一段线，它是电阻，我们给它加上电势差V。现在，V的意思就是，如果我们把一个电荷，从线的一端，载到另外一端，那么，所做的功，就是qV。当电荷从高电压处，“掉到”低电压处时，电压差越高，所做的功就越多，于是，在电荷从一端，走到另外一端的过程中，会释放能量。现在，电核并不是简单地从一端，飞到另一端；线中的原子，给电流带来了一定阻力，这个阻力，遵守如下规律，几乎对于所有普通物质，都成立：如果有电流 I，也就是说，每秒经过线的，有如此如此这么多的电荷；每秒经过线的电荷数目，就正比于，我们推它们有多难--换句话说，正比于那里的电压是多少：
V=IR=R(dq/dt). (25.11)
The coefficient R is called the resistance, and the equation is called Ohm’s Law.The unit of resistance is the ohm; it is equal to one volt per ampere. In mechanicalsituations, to get such a frictional force in proportion to the velocity is difficult;in an electrical system it is very easy, and this law is extremely accurate formost metals.
系数R，被称为电阻，此方程被称为欧姆规律。电阻的单位，就是欧姆；它等于：一伏特每安培。在机械情况中，要得到，这样一种正比于矢速的摩擦力，是困难的；在电子系统中，则很容易，这个规律，对于大多数金属，极为精确。

We are often interested in how much work isdone per second, the power loss, or the energy liberated by the charges as theytumble down the wire. When we carry a charge q through a voltage V , the work is qV , so the work done per second would be V(dq/dt), which is the same as VI , or also IR⋅I=I2R . This is called the heating loss—this is how much heat isgenerated in the resistance per second, by the conservation of energy. It is thisheat that makes an ordinary incandescent light bulb work.
我们通常感兴趣的，是每秒做了多少功、功率的损失是多少、或当电荷翻滚过电线时，释放的能量是多少。当我们把一个电荷q，载过一个电压时V，所做的功就是qV，于是，每秒所做的功就是V(dq/dt)，它与VI一样，也可写作 IR⋅I=I2R，这被称为加热损失，这就是电阻中，每秒产生了多少热，通过能量守恒得到的。让普通白炽灯发光的，就是这个热。

Now what electrical thing corresponds tothe mechanical spring, in which there was a force proportional to the stretch?If we start with F=kx and replace F→V and x→q , we get V=αq . It turns out that there is such a thing, in fact it is theonly one of the three circuit elements we can really understand, because we didstudy a pair of parallel plates, and we found that if there were a charge ofcertain equal, opposite amounts on each plate, the electric field between themwould be proportional to the size of the charge. So the work done in moving aunit charge across the gap from one plate to the other is precisely proportionalto the charge. This work is the definition of the voltage difference,and it is the line integral of the electric field from one plate to another. Itturns out, for historical reasons, that the constant of proportionality is notcalled C , but 1/C . It could have been called C , but it was not. So we have
弹簧中的力，正比于其拉伸，那么，在电学中，与弹簧相应的，是什么呢？如果我们从F=kx开始，做替换F→V 和 x→q ，我们就得到V=αq。结果就是，确实有这么一个事情，事实上，在电路的三个要素中，它是唯一一个我们可以真正理解的，因为，我们确实研究了一对平行板，且我们发现，如果在每个板子上，都有一定量的电荷，量相等且性相反，那么，它们之间的电场，就会正比于电荷的多少。所以，把一个单位的电荷，从一个板，移到另外一个板，所做的功，精确地正比于电荷。这个功，就是电压差的定义，它就是电场的线性积分，从一个板到另一个板的。结果就是，由于历史的原因，正比的常数，不是被称为C , 而是 1/C。它可以被称为C ，但是没有。所以，我们就有：
V=q/C. (25.13)
The unit of capacitance, C , is the farad; a charge of one coulomb on each plate of a one-faradcapacitor yields a voltage difference of one volt.
电容C的单位，是法拉第；在一法拉第的电容器上，一库伦的电量，能产生一伏特的电压差。