The History of Mathematical Physics
The Past and the Future of Physics.— What is the present state of mathematical physics? What are the problems it is led to set itself? What is its future? Is its orientation1 about to be modified?
Ten years hence will the aim and the methods of this science appear to our immediate2 successors in the same light as to ourselves; or, on the contrary, are we about to witness a profound transformation3? Such are the questions we are forced to raise in entering to-day upon our investigation4.
If it is easy to propound5 them: to answer is difficult. If we felt tempted6 to risk a prediction, we should easily resist this temptation, by thinking of all the stupidities the most eminent7 savants of a hundred years ago would have uttered, if some one had asked them what the science of the nineteenth century would be. They would have thought themselves bold in their predictions, and after the event, how very timid we should have found them. Do not, therefore, expect of me any prophecy.
But if, like all prudent8 physicians, I shun9 giving a prognosis, yet I can not dispense10 with a little diagnostic; well, yes, there are indications of a serious crisis, as if we might expect an approaching transformation. Still, be not too anxious: we are sure the patient will not die of it, and we may even hope that this crisis will be salutary, for the history of the past seems to guarantee us this. This crisis, in fact, is not the first, and to understand it, it is important to recall those which have preceded. Pardon then a brief historical sketch11.
The Physics of Central Forces.— Mathematical physics, as we know, was born of celestial12 mechanics, which gave birth to it at the end of the eighteenth century, at the moment when it itself attained13 its complete development. During its first years especially, the infant strikingly resembled its mother.
The astronomic15 universe is formed of masses, very great, no doubt, but separated by intervals16 so immense that they appear to us only as material points. These points attract each other inversely18 as the square of the distance, and this attraction is the sole force which influences their movements. But if our senses were sufficiently19 keen to show us all the details of the bodies which the physicist20 studies, the spectacle thus disclosed would scarcely differ from the one the astronomer21 contemplates22. There also we should see material points, separated from one another by intervals, enormous in comparison with their dimensions, and describing orbits according to regular laws. These infinitesimal stars are the atoms. Like the stars proper, they attract or repel23 each other, and this attraction or this repulsion, following the straight line which joins them, depends only on the distance. The law according to which this force varies as function of the distance is perhaps not the law of Newton, but it is an analogous24 law; in place of the exponent25 ?2, we have probably a different exponent, and it is from this change of exponent that arises all the diversity of physical phenomena26, the variety of qualities and of sensations, all the world, colored and sonorous27, which surrounds us; in a word, all nature.
Such is the primitive28 conception in all its purity. It only remains29 to seek in the different cases what value should be given to this exponent in order to explain all the facts. It is on this model that Laplace, for example, constructed his beautiful theory of capillarity; he regards it only as a particular case of attraction, or, as he says, of universal gravitation, and no one is astonished to find it in the middle of one of the five volumes of the ‘Mécanique céleste.’ More recently Briot believes he penetrated31 the final secret of optics in demonstrating that the atoms of ether attract each other in the inverse17 ratio of the sixth power of the distance; and Maxwell himself, does he not say somewhere that the atoms of gases repel each other in the inverse ratio of the fifth power of the distance? We have the exponent ?6, or ?5, in place of the exponent ?2, but it is always an exponent.
Among the theories of this epoch32, one alone is an exception, that of Fourier; in it are indeed atoms acting33 at a distance one upon the other; they mutually transmit heat, but they do not attract, they never budge34. From this point of view, Fourier’s theory must have appeared to the eyes of his contemporaries, to those of Fourier himself, as imperfect and provisional.
This conception was not without grandeur35; it was seductive, and many among us have not finally renounced36 it; they know that one will attain14 the ultimate elements of things only by patiently disentangling the complicated skein that our senses give us; that it is necessary to advance step by step, neglecting no intermediary; that our fathers were wrong in wishing to skip stations; but they believe that when one shall have arrived at these ultimate elements, there again will be found the majestic37 simplicity38 of celestial mechanics.
Neither has this conception been useless; it has rendered us an inestimable service, since it has contributed to make precise the fundamental notion of the physical law.
I will explain myself; how did the ancients understand law? It was for them an internal harmony, static, so to say, and immutable39; or else it was like a model that nature tried to imitate. For us a law is something quite different; it is a constant relation between the phenomenon of to-day and that of to-morrow; in a word, it is a differential equation.
Behold40 the ideal form of physical law; well, it is Newton’s law which first clothed it forth41. If then one has acclimated42 this form in physics, it is precisely43 by copying as far as possible this law of Newton, that is by imitating celestial mechanics. This is, moreover, the idea I have tried to bring out in Chapter VI.
The Physics of the Principles.— Nevertheless, a day arrived when the conception of central forces no longer appeared sufficient, and this is the first of those crises of which I just now spoke44.
What was done then? The attempt to penetrate30 into the detail of the structure of the universe, to isolate45 the pieces of this vast mechanism46, to analyze47 one by one the forces which put them in motion, was abandoned, and we were content to take as guides certain general principles the express object of which is to spare us this minute study. How so? Suppose we have before us any machine; the initial wheel work and the final wheel work alone are visible, but the transmission, the intermediary machinery48 by which the movement is communicated from one to the other, is hidden in the interior and escapes our view; we do not know whether the communication is made by gearing or by belts, by connecting-rods or by other contrivances. Do we say that it is impossible for us to understand anything about this machine so long as we are not permitted to take it to pieces? You know well we do not, and that the principle of the conservation of energy suffices to determine for us the most interesting point. We easily ascertain49 that the final wheel turns ten times less quickly than the initial wheel, since these two wheels are visible; we are able thence to conclude that a couple applied50 to the one will be balanced by a couple ten times greater applied to the other. For that there is no need to penetrate the mechanism of this equilibrium51 and to know how the forces compensate52 each other in the interior of the machine; it suffices to be assured that this compensation can not fail to occur.
Well, in regard to the universe, the principle of the conservation of energy is able to render us the same service. The universe is also a machine, much more complicated than all those of industry, of which almost all the parts are profoundly hidden from us; but in observing the motion of those that we can see, we are able, by the aid of this principle, to draw conclusions which remain true whatever may be the details of the invisible mechanism which animates53 them.
The principle of the conservation of energy, or Mayer’s principle, is certainly the most important, but it is not the only one; there are others from which we can derive54 the same advantage. These are:
Carnot’s principle, or the principle of the degradation55 of energy.
Newton’s principle, or the principle of the equality of action and reaction.
The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary56 observer as for an observer carried along in a uniform motion of translation; so that we have not and can not have any means of discerning whether or not we are carried along in such a motion.
The principle of the conservation of mass, or Lavoisier’s principle.
I will add the principle of least action.
The application of these five or six general principles to the different physical phenomena is sufficient for our learning of them all that we could reasonably hope to know of them. The most remarkable57 example of this new mathematical physics is, beyond question, Maxwell’s electromagnetic theory of light.
We know nothing as to what the ether is, how its molecules58 are disposed, whether they attract or repel each other; but we know that this medium transmits at the same time the optical perturbations and the electrical perturbations; we know that this transmission must take place in conformity59 with the general principles of mechanics, and that suffices us for the establishment of the equations of the electromagnetic field.
These principles are results of experiments boldly generalized; but they seem to derive from their very generality a high degree of certainty. In fact, the more general they are, the more frequent are the opportunities to check them, and the verifications multiplying, taking the most varied60, the most unexpected forms, end by no longer leaving place for doubt.
Utility of the Old Physics.— Such is the second phase of the history of mathematical physics and we have not yet emerged from it. Shall we say that the first has been useless? that during fifty years science went the wrong way, and that there is nothing left but to forget so many accumulated efforts that a vicious conception condemned61 in advance to failure? Not the least in the world. Do you think the second phase could have come into existence without the first? The hypothesis of central forces contained all the principles; it involved them as necessary consequences; it involved both the conservation of energy and that of masses, and the equality of action and reaction, and the law of least action, which appeared, it is true, not as experimental truths, but as theorems; the enunciation62 of which had at the same time something more precise and less general than under their present form.
It is the mathematical physics of our fathers which has familiarized us little by little with these various principles; which has habituated us to recognize them under the different vestments in which they disguise themselves. They have been compared with the data of experience, it has been seen how it was necessary to modify their enunciation to adapt them to these data; thereby63 they have been extended and consolidated64. Thus they came to be regarded as experimental truths; the conception of central forces became then a useless support, or rather an embarrassment65, since it made the principles partake of its hypothetical character.
The frames then have not broken, because they are elastic66; but they have enlarged; our fathers, who established them, did not labor67 in vain, and we recognize in the science of to-day the general traits of the sketch which they traced.
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1
orientation
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| n.方向，目标；熟悉，适应，情况介绍 | |
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immediate
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| adj.立即的；直接的，最接近的；紧靠的 | |
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transformation
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| n.变化；改造；转变 | |
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investigation
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| n.调查，调查研究 | |
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propound
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| v.提出 | |
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tempted
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| v.怂恿（某人）干不正当的事；冒…的险（tempt的过去分词） | |
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eminent
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| adj.显赫的，杰出的，有名的，优良的 | |
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prudent
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| adj.谨慎的，有远见的，精打细算的 | |
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shun
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| vt.避开，回避，避免 | |
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dispense
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| vt.分配，分发；配(药)，发(药)；实施 | |
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sketch
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| n.草图；梗概；素描；v.素描；概述 | |
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12
celestial
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| adj.天体的；天上的 | |
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attained
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| （通常经过努力）实现( attain的过去式和过去分词 )； 达到； 获得； 达到（某年龄、水平、状况） | |
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attain
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| vt.达到，获得，完成 | |
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astronomic
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| 天文学的，星学的 | |
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intervals
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| n.[军事]间隔( interval的名词复数 )；间隔时间；[数学]区间；（戏剧、电影或音乐会的）幕间休息 | |
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inverse
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| adj.相反的，倒转的，反转的；n.相反之物；v.倒转 | |
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inversely
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| adj.相反的 | |
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sufficiently
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| adv.足够地，充分地 | |
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physicist
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| n.物理学家，研究物理学的人 | |
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astronomer
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| n.天文学家 | |
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contemplates
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| 深思，细想，仔细考虑( contemplate的第三人称单数 )； 注视，凝视； 考虑接受（发生某事的可能性）； 深思熟虑，沉思，苦思冥想 | |
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repel
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| v.击退，抵制，拒绝，排斥 | |
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analogous
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| adj.相似的；类似的 | |
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exponent
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| n.倡导者，拥护者；代表人物；指数，幂 | |
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phenomena
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| n.现象 | |
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sonorous
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| adj.响亮的，回响的；adv.圆润低沉地；感人地；n.感人，堂皇 | |
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primitive
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| adj.原始的；简单的；n.原(始)人，原始事物 | |
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remains
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| n.剩余物，残留物；遗体，遗迹 | |
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penetrate
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| v.透(渗)入；刺入，刺穿；洞察，了解 | |
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penetrated
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| adj. 击穿的，鞭辟入里的 动词penetrate的过去式和过去分词形式 | |
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epoch
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| n.(新)时代；历元 | |
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acting
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| n.演戏，行为，假装；adj.代理的，临时的，演出用的 | |
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budge
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| v.移动一点儿；改变立场 | |
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grandeur
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| n.伟大，崇高，宏伟，庄严，豪华 | |
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renounced
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| v.声明放弃( renounce的过去式和过去分词 )；宣布放弃；宣布与…决裂；宣布摒弃 | |
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majestic
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| adj.雄伟的，壮丽的，庄严的，威严的，崇高的 | |
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simplicity
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| n.简单，简易；朴素；直率，单纯 | |
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immutable
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| adj.不可改变的，永恒的 | |
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behold
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| v.看，注视，看到 | |
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forth
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| adv.向前；向外，往外 | |
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acclimated
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| v.使适应新环境，使服水土服水土，适应( acclimate的过去式和过去分词 ) | |
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precisely
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| adv.恰好，正好，精确地，细致地 | |
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44
spoke
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| n.（车轮的）辐条；轮辐；破坏某人的计划；阻挠某人的行动 v.讲，谈（speak的过去式）；说；演说；从某种观点来说 | |
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45
isolate
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| vt.使孤立，隔离 | |
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mechanism
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| n.机械装置；机构，结构 | |
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analyze
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| vt.分析，解析 (=analyse) | |
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machinery
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| n.(总称)机械，机器；机构 | |
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49
ascertain
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| vt.发现，确定，查明，弄清 | |
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applied
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| adj.应用的；v.应用，适用 | |
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equilibrium
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| n.平衡，均衡，相称，均势，平静 | |
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compensate
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| vt.补偿，赔偿；酬报 vi.弥补；补偿；抵消 | |
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53
animates
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| v.使有生气( animate的第三人称单数 )；驱动；使栩栩如生地动作；赋予…以生命 | |
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derive
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| v.取得；导出；引申；来自；源自；出自 | |
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degradation
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| n.降级；低落；退化；陵削；降解；衰变 | |
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stationary
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| adj.固定的，静止不动的 | |
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remarkable
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| adj.显著的，异常的，非凡的，值得注意的 | |
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molecules
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| 分子( molecule的名词复数 ) | |
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conformity
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| n.一致，遵从，顺从 | |
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varied
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| adj.多样的，多变化的 | |
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condemned
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| adj. 被责难的, 被宣告有罪的 动词condemn的过去式和过去分词 | |
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enunciation
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| n.清晰的发音；表明，宣言；口齿 | |
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thereby
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| adv.因此，从而 | |
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consolidated
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| a.联合的 | |
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embarrassment
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| n.尴尬；使人为难的人(事物)；障碍；窘迫 | |
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66
elastic
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| n.橡皮圈，松紧带；adj.有弹性的；灵活的 | |
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labor
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| n.劳动，努力，工作，劳工；分娩；vi.劳动，努力，苦干；vt.详细分析；麻烦 | |
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