Of course the solution we have found is thesolution only if things are started just right, for otherwise there is a partwhich usually dies out after a while. This other part is called the transientresponse to F(t) , while (21.10)and (21.12)are called the steady-state response.
当然，我们刚发现的这个解，是事情正确启动时的解，否则的话，就会有一部分，通常过一会儿，就会死掉。这部分，被称为对 F(t)的短暂反应，而（20.10）和（20.12），则被称为稳定状态反应。

According to our formula (21.12),a very remarkable thing should also occur: if ω is almost exactly the same as ω0 , then C should approach infinity. So if we adjust the frequency of the forceto be “in time” with the natural frequency, then we should get an enormousdisplacement. This is well known to anybody who has pushed a child on a swing.It does not work very well to close our eyes and push at a certain speed atrandom. If we happen to get the right timing, then the swing goes very high,but if we have the wrong timing, then sometimes we may be pushing when weshould be pulling, and so on, and it does not work.
依据我们的公式（21.12），就会发生一个非常值得注意的事情：如果ω与ω0，几乎一样，那么，C将接近无穷大。于是，如果我们调整力的频率，让它与自然频率“合拍”，那么，我们将会得到一个巨大的位移，任何人，如果推过小孩子荡秋千，都会知道这一点。如果我们闭着眼睛，随机地推，效果不大。如果我们掌握了正确的时机，那么，秋千就会荡的很高，但是，如果我们时机不对，该拉时，却是在推，等等，那么，秋千难荡。

If we make ω exactly equal to ω0 , we find that it should oscillate at an infinite amplitude,which is, of course, impossible. The reason it does not is that something goeswrong with the equation, there are some other frictional terms, and otherforces, which are not in (21.8)but which occur in the real world. So the amplitude does not reach infinity forsome reason; it may be that the spring breaks!
如果我们让ω准确地等于 ω0，我们发现，振幅将会是无限的，这当然是不可能的。之所以不能，乃是因为，方程有错，有些其他的摩擦项，其他的力，并没有在（21.8）中，但在现实世界中，它们发生了。所以，由于某些原因，振幅达不到无限；比如说，弹簧断了。

Chapter22 Algebra 第22章 代数
22–1Addition and multiplication 22-1 加法与乘法
In our study of oscillating systems weshall have occasion to use one of the most remarkable, almost astounding,formulas in all of mathematics. From the physicist’s point of view we couldbring forth this formula in two minutes or so, and be done with it. But scienceis as much for intellectual enjoyment as for practical utility, so instead ofjust spending a few minutes on this amazing jewel, we shall surround the jewelby its proper setting in the grand design of that branch of mathematics whichis called elementary algebra.
在我们对振荡系统的研究中，我们将有机会使用，所有数学中最著名的、或者也是令人震惊的公式。从物理学家的观点，我们可以在一、两分钟内，就把这个公式，拿出来，且能理解应用。但是，科学对于智力享受，与对实际使用，是同样地多，所以，对于这个令人惊奇的瑰宝，除了花几分钟时间处理外，我们还要通过其合适背景的伟大设计，来环视这个瑰宝，这里背景，是指基础代数，它是数学的一个分支。{？}

Now you may ask, “What is mathematics doingin a physics lecture?” We have several possible excuses: first, of course,mathematics is an important tool, but that would only excuse us for giving theformula in two minutes. On the other hand, in theoretical physics we discoverthat all our laws can be written in mathematical form; and that this has acertain simplicity and beauty about it. So, ultimately, in order to understandnature it may be necessary to have a deeper understanding of mathematicalrelationships. But the real reason is that the subject is enjoyable, and althoughwe humans cut nature up in different ways, and we have different courses indifferent departments, such compartmentalization is really artificial, and we shouldtake our intellectual pleasures where we find them.
现在，我们可以问：“在一个物理讲座中，数学有什么用呢？”我们有几种可能的借口：第一，当然，数学是一个重要的工具，但是，这只能替我们解释，在两分钟内拿出公式。另一方面，在理论物理学中，我们发现，我们所有的规律，都可以用数学的形式来写，这样，就会让物理，具有一定的简明和漂亮。所以，最终，为了理解自然，就有必要，对数学关系，有更深的理解。但是，真正的原因则是，主题很有乐趣，虽然我们人类，以不同的方式，划分自然，我们在不同的系，有不同的课程，但这种组织划分，真正是人工的；对于智力快乐，我们在哪里发现了它，我们就应该在那里去享受它。

Another reason for looking more carefullyat algebra now, even though most of us studied algebra in high school, is thatthat was the first time we studied it; all the equations were unfamiliar, andit was hard work, just as physics is now. Every so often it is a great pleasureto look back to see what territory has been covered, and what the great map orplan of the whole thing is. Perhaps some day somebody in the Mathematics Departmentwill present a lecture on mechanics in such a way as to show what it was we weretrying to learn in the physics course!
现在，要更仔细地查看代数，还有另外一个原因，虽然，我们大多数人，在高中都学过代数，但那是我们第一次学它；所有的公式，都不熟悉，那个过程很艰苦，正如现在的物理一样。经常回头看看，都有哪些领域，被涵盖了，整个事情的伟大的路线图或计划，又是什么，会有很大乐趣。或许有一天，数学系的某人，会以如下方式，提供一个关于力学的讲座：即指出，我们过去，在物理学中，所学的东西，究竟是什么！

The subject of algebra will not bedeveloped from the point of view of a mathematician, exactly, because the mathematiciansare mainly interested in how various mathematical facts are demonstrated, andhow many assumptions are absolutely required, and what is not required. Theyare not so interested in the result of what they prove. For example, we mayfind the Pythagorean theorem quite interesting, that the sum of the squares ofthe sides of a right triangle is equal to the square of the hypotenuse; that isan interesting fact, a curiously simple thing, which may be appreciated withoutdiscussing the question of how to prove it, or what axioms are required. So, inthe same spirit, we shall describe qualitatively, if we may put it that way,the system of elementary algebra. We say elementary algebra becausethere is a branch of mathematics called modern algebra in which some ofthe rules such as ab=ba , are abandoned, and it is still called algebra, but we shall notdiscuss that.
代数这一主题，从数学家的观点看，准确地说，并不会被发展，因为，数学家们主要感兴趣的，是各种不同的数学事实，是如何被演证的，以及有多少假设，被绝对地要求了，及什么，并没有被要求。对于他们所证明的结果，他们并不是特别感兴趣。例如，我们可以发现，毕达哥拉斯定理，非常有趣，一个直角三角形，两直边的平方和，等于斜边的平方；这是一个有趣的事实，一件稀奇而简明的事情，无需讨论如何证明它、及需要什么公理，我们也可以赞赏它。所以，以同样的精神，我们将定性地描述基本代数，如果我们可以这样说的话。我们说基本代数，乃是因为，还有一个数学分支，被称为现代代数，在那里，有些规则，例如ab=ba，是被禁止的，它仍被称为代数，但是，我们不会讨论它。

To discuss this subject we start in themiddle. We suppose that we already know what integers are, what zero is, andwhat it means to increase a number by one unit. You may say, “That is not inthe middle!” But it is the middle from a mathematical standpoint, because wecould go even further back and describe the theory of sets in order to derivesome of these properties of integers. But we are not going in that direction,the direction of mathematical philosophy and mathematical logic, but rather inthe other direction, from the assumption that we know what integers are and weknow how to count.
要讨论这个主题，我们从中间开始。我们假设，我们已经知道了，整数是什么，零是什么，以及，把一个数，按单位来增加，意味着什么。你可以说：“这并不是在中间！”但是，从数学的观点看，它是中间，因为，我们不可能往回走的更远，为了导出一些整数的属性，而去描述集合理论。但是，我们不会往那个方向走，那是数学哲学和数学逻辑的方向，我们宁肯往其他的方向走，即假设，我们已经知道了，整数是什么，及如何计数。

If we start with a certain number a, an integer, and we count successively one unit b times, the number we arrive at we call a+b , and that defines addition of integers.
设有整数 a，如果我们从它开始，对于同一个单位，我们成功地计算了b次，那么，我们达到的数，我们称之为a+b，这样，就定义了整数的加法。

Once we have defined addition, then we canconsider this: if we start with nothing and add a to it, b times in succession, we call the result multiplication ofintegers; we call it b times a .
一旦我们定义了加法，那么，我们就可以这样考虑：如果我们从零开始，先把a加给它，共b次，我们把此结果，称为整数的乘法；我们称之为b 乘以 a。

Now we can also have a succession of multiplications:if we start with 1 and multiply by a , b times in succession, we call that raising to a power: ab.
现在，我们还可以有乘法的连续：如果我们从1开始，让它乘以 a，共b次，我们称之为次方：ab 。

Now as a consequence of these definitionsit can be easily shown that all of the following relationships are true:
现在，作为这些定义的后果，很易指出，下面所有关系，都为真：
(22.1)

These results are well known and we shall not belabor the point, wemerely list them. Of course, 1 and 0 have special properties; for example, a+0 is a , a times 1=a , and a to the first power is a .
这些结果，广为人知，恕不赘述，只是列出。当然，1和0，具有特别属性；例如，a+0 是 a , a 乘以1=a , a 的一次方是 a 。

In this discussion we must also assume a fewother properties like continuity and ordering, which are very hard to define;we will let the rigorous theory do it. Furthermore, it is definitely true thatwe have written down too many “rules”; some of them may be deducible from theothers, but we shall not worry about such matters.
在这个讨论中，我们还应假定一些其他的属性，比如连续性和排列组合，这些都很难定义；我们将让严格细致的理论，去做它。另外，完全真实的是，我们已经写下了太多的“规则”；其中一些，可以从另一些推出，但是，对此，我们无需担忧。

22–2The inverse operations 22-2 逆运算
In addition to the direct operations ofaddition, multiplication, and raising to a power, we have also the inverseoperations, which are defined as follows. Let us assume that a and c are given, and that we wish to find what values of b satisfy such equations as a+b=c , ab=c , ba=c . If a+b=c , b is defined as c−a , which is called subtraction. The operation called division isalso clear: if ab=c , then b=c/a defines division—a solution of the equation ab=c “backwards.” Now if we have a power ba=c and we ask ourselves, “What is b ?,” it is called the ath root of c : b=（c）1/a . For instance, if we ask ourselves the following question, “Whatinteger, raised to the third power, equals 8 ?,” then the answer is called the cube root of 8 ; it is 2 . Because ba and ab are not equal, there are two inverse problems associated withpowers, and the other inverse problem would be, “To what power must weraise 2 to get 8 ?” This is called taking the logarithm. If ab=c, we write b=logac . The fact that it has a cumbersome notation relative to the othersdoes not mean that it is any less elementary, at least applied to integers,than the other processes. Although logarithms come late in an algebra class, inpractice they are, of course, just as simple as roots; they are just adifferent kind of solution of an algebraic equation. The direct and inverseoperations are summarized as follows:
除了直接的加法、乘法、和求次方外，我们还有逆运算，它们定义如下。设a和c被给予了，我们希望找到一个b，其值满足方程a+b=c , ab=c , ba=c 。如果 a+b=c ，那么， b就被定义为c−a ，这被称为减法。被称为除法的运算，也清楚了，如果ab=c , 那么，b=c/a就定义了除法，它是方程ab=c的“反向”。现在，如果我们有一个次方ba=c，我们要问“b是多少？”它被称为c的a次方根：b=（c）1/a 。例如，如果我们问自己如下问题：“什么整数的三次方等于8？”那么，答案就是，8的立方根；它就是2。因为ba与 ab，并不相等，所以，对于次方，就有两个逆运算，另外一个逆运算的问题就是：“2的几次方等于8？”这被称为取对数。如果ab=c , 我们写b=logac。与其他表示法相比，对数表示法，比较笨重，但此事实，并不意味着，与其他运算过程相比，它缺少基本运算的性质，至少，是可以应用于整数的。虽然在代数中，对数来的比较晚，但是在实践中，当然，它与求根，同样简明；对于同一个代数方程，它们只不过是其不同解。直接运算和逆向运算，总结如下：
(22.2)