数学物理原理

Principles of Mathematical Physics Author(s): Henri Poincaré Source: The Scientific Monthly, Vol. 82, No. 4 (Apr., 1956), pp. 165-175 Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/21942 Accessed: 25/07/2009 09:50 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=aaas. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. American Association for the Advancement of Science is collaborating with JSTOR to digitize, preserve and extend access to The Scientific Monthly. http://www.jstor.org SCIENTIFIC MONTHLY APRIL 1956 Principles of Mathematical Physics HENRI POINCARE At the beginning of the present century could anyone have predicted the striking developments in physics soon to take place? One man came surprisingly close. He was Henri Poincare (1854-1912), professor at the Sorbonne in Paris. In 1904, a year before the publication by Einstein of his fundamental papers on special relativity, quantum theory, and the Brownian movement, Poincare' examined the state of physical research and offered to the International Con- gress of Arts and Sciences in St. Louis what he jokingly termed his diagnosis. Foresight is a matter of genius, hindsight is something all can enjoy. On the following pages, available for comparison with what has actually happened, are Poincare"s remarkable predictions of more than half a century ago. Poin- care"s own contribution to science was broad and deep, enriching pure mathe- matics, mathematical physics, and astronomy. His writings on the meaning and methods of science are now accepted as classics of philosophy. The St. Louis address on "The principles of mathematical physics," translated by George Bruce Halsted, was published in The Monist, January 1905; it served subse- quently as the basis for several chapters in Poincare5's The Value of Science. W H HAT is the actual state of mathemati- cal physics? What are the problems it is led to set itself? What is its future? Is its orientation on the point of modifying itself? Will the aim and the methods of this science ap- pear in ten years to our immediate successors in the same light as to ourselves; or, on the contrary, are we about to witness a profound transformation? Such are the questions we are forced to raise in entering today upon our investigation. If it is easy to propound them, to answer is diffi- cult. If we feel ourselves tempted to risk a prognosti- cation, we have, to resist this temptation, only to think of all the stupidities the most eminent sa- vants of a hundred years agro would have uttered, if one had asked them what the science of the nine- teenth century would be. They would have believed 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; if I had known what one will discover tomorrow, I would long ago have published it to secure me the priority. But if, like all prudent physicians, I shun giving a prognosis, nevertheless I cannot dispense with a little diagnostic; well, yes, there are indications of a serious crisis, as if we should expect an ap- proaching transformation. We are assured that the patient will not die of it, and even we can hope that this crisis will be salu- tary, that it was even necessary for his develop- ment. This the history of the past seems to guaran- tee us. This crisis in fact is not the first, and for its com- prehension it is important to recall those which have preceded it. Mathematical physics, we know, was born of celestial mechanics, which engendered it at the end of the eighteenth century, at the moment when it itself attained its complete development. During its first years especially, the infant resembled in a striking way its mother. The astronomic universe is formed of masses, April 1956 165 very great without doubt, but separated by inter- vals so immense, that they appear to us only as material points. These points attract each other in the inverse ratio of the square of the distances, and this attraction is the sole force which influences their movements. But if our senses were sufficiently subtle to show us all the details of the bodies which the physicist studies, the spectacle we should there discover would scarcely differ from what the as- tronomer contemplates. There also we should see material points, separated one from another by in- tervals, enormous in relation to their dimensions, and describing orbits following regular laws. These infinitesimal stars are the atoms. Like the stars properly so called, they attract or repel each other, and this attraction or this repulsion, directed following the straight line which joins them, de- pends only on the distance. The law according to which this force varies as a function of the distance is perhaps not the law of Newton, but it is an anal- ogous law; in place of the exponent - 2, we prob- ably have a different exponent, and it is from this change of exponent that springs all the diversity of physical phenomena, the variety of qualities and of sensations, all the world colored and sonorous which surrounds us in a word, all nature. Such is the primitive conception in all its purity. It only remains 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 particu- lar 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 Mecanique celeste. More recently Briot believed he had penetrated the final secret of optics in demonstrating that the atoms of ether attract each other in the inverse ratio of the sixth power of the distance; and Max- well, Maxwell himself, does he not say somewhere that the atoms of gases repel each other in the in- verse ratio of the fifth power of the distance? We have the exponent - 6, or - 5 in place of the expo- nent-2, but it is always an exponent. Among the theories of this epoch, one alone is an exception, that of Fourier; in it are indeed atoms, acting at a distance one upon the other; they mu- tually transmit heat, but they do not attract, they never budge. From this point of view, the theory of Fourier must have appeared to the eyes of his contemporaries, to those of Fourier himself, as im- perfect and provisional. This conception was not without grandeur; it was seductive, and many among us have not finally renounced it; they know that one will attain the ultimate elements of things only by patiently dis- entangling 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 majestic simplicity of celestial mechanics. Neither has this conception been useless; it has rendered us an inestimable service, since it has con- tributed to make precise in us the fundamental notion of the physical law. I will explain myself; how did the ancients un- derstand law? It was for them an internal har- mony, static, so to say, and immutable; or it was like a model that nature constrained herself to imitate. A law for us is no more that at all; it is a constant relation between the phenomenon of to- day and that of tomorrow; in a word, it is a differ- ential equation. Behold the ideal form of physical law; well, it is the law of Newton which first covered it; and then how has one acclimated this form in physics; pre- cisely in copying as much as possible this law of Newton-that is, in imitating celestial mechanics. 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 spoke. What did one do then? One gave up trying to penetrate into the detail of the structure of the universe, to isolate the pieces of this vast mecha- nism, to analyze one by one the forces which put them in motion, and was content to take as guides certain general principles which have precisely for object to spare us this minute study. How so? Suppose that we have before us any machine; the initial wheel work and the final wheel work alone are visible, but the transmission, the intermediary wheels by which the movement is communicated from one to the other are hidden in the interior and escape our view; we do not know whether the communication is made by gear- ing or by belts, by connecting rods or by other dispositives. Do we say that it is impossible for us to under- stand 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 con- servation of energy suffices to determine for us the most interesting point. We easily ascertain 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 ap- 166 THE SCIENTIFIC MONTHLY plied 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 equi- librium and to know how the forces compensate each other in the interior of the machine; it suffices to be assured that this compensation cannot fail to occur. Well, in regard to the universe, the principle of the conservation of energy is able to render us the same service. This is also a machine, much more complicated than all those of industry, and of which almost all the parts are profoundly hidden from us; but in observing the movement of those that we can see, we are able, aiding ourselves by this principle, to draw conclusions which remain true whatever may be the details of the invisible mechanism which animates them. The principle of the conservation of energy, or the principle of Mayer, is certainly the most impor- tant, but it is not the only one; there are others from which we are able to draw the same advan- tage. These are the principle of Carnot, or the principle of the degradation of energy; the princi- ple of Newton, or the principle of the equality of action and reaction; the principle of relativity, ac- cording to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform move- ment of translation; so that we have not and could 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 principle of Lavoi- sier. I would add the principle of least action. The application of these five or six general prin- ciples to the different physical phenomena is suffi- cient for our learning of them what we could reasonably hope to know of them. The most remarkable example of this new math- ematical physics is, beyond contradiction, Max- well's electromagnetic theory of light. We know nothing as to what is the ether, how its molecules 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 should be made conformable to 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 generality itself an eminent degree of certitude. In fact, the more general they are, the more frequently one has the occasion to check them, and the verifications, in multiplying themselves, in taking forms the most varied and the most unex- pected, finish by leaving no longer place for doubt. Such is the second phase of the history of mathe- matical physics and we have not yet emerged from it. Do 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 con- demned in advance to nonsuccess? Not the least in the world. Do you believe that 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 con- sequences; it involved both the conservation of energy and that of masses, and the equality of ac- tion and reaction; and the law of least action, which would appear, it is true, not as experimental verities, but as theorems and of which the enuncia- tion would have at the same time a something more precise and less general than under their actual form. It is the mathematical physics of our fathers which has familiarized us little by little with these divers principles; which has habituated us to recog- nize them under the different vestments in which they disguise themselves. One has compared them to the data of experience, or has seen how it was necessary to modify their enunciation to adapt them to these data; thereby they have been en- larged and consolidated. So one has been led to regard them as experi- mental verities; the conception of central forces became then a useless support, or rather an embar- rassment, since it made the principles partake of its hypothetical character. The frames have not therefore broken, because they were elastic; but they have enlarged; our fathers, who established them, did not work in vain, and we recognize in the science of today the gen- eral traits of the sketch which they traced. Are we about to enter now upon the eve of a sec- ond crisis? These principles on which we have built all are they about to crumble away in their turn? Since some time, this may well be asked. In hearing me speak thus, you think without doubt of radium, that grand revolutionist of the present time, and in fact I will come back to it presently; but there is something else. It is not alone the conservation of energy which is in question; all the other principles are equally in danger, as we shall see in passing them succes- sively in review. Let us commence with the principle of Carnot. April 1956 167 This is the only one which does not present itself as an immediate consequence of the hypothesis of central forces; more than that, it seems, if not to directly contradict that hypothesis, at least not to be reconciled with it without a certain effort. If physical phenomena were due exclusively to the movements of atoms whose mutual attraction depended only on the distance, it seems that all these phenomena should be reversible; if all the initial velocities were reversed, these atoms, always subjected to the same forces, ought to go over their trajectories in the contrary sense, just as the earth would describe in the retrograde sense this same elliptic orbit which it describes in the direct sense, if the initial conditions of its movement had been reversed. On this account, if a physical phenom- enon is possible, the inverse phenomenon should be equally so, and one should be able to reascend the course of time. But it is not so in nature, and this is precisely what the principle of Carnot teaches us; heat can pass from the warm body to the cold body; it is im- possible afterwards to make it reascend the inverse way and reestablish differences of temperature which have been effaced. Motion can be wholly dissipated and trans- formed into heat by friction; the contrary trans- formation can never be made except in a partial manner. We have striven to reconcile this apparent con- tradiction. If the world tends toward uniformity, this is not because its ultimate parts, at first unlike, tend to become less and less different; it is because, shifting at hazard, they end by blending. For an eye which should distinguish all the elements, the variety would remain always as great, each grain of this dust preserves its originality and does not model itself on its neighbors; but as the blend be- comes more and more intimate, our gross senses perceive no more than the uniformity. Behold why, for example, temperatures tend to a level, without the possibility of turning backwards. A drop of wine falls into a glass of water; what- ever may be the law of the internal movements of the liquid, we soon see it colored of a uniform rosy tint and from this moment, one may well shake the vase, the wine and the water do not seem able any more to separate. See, thus, what would be the type of the reversible physical phenomenon: to hide a grain of barley in a cup of wheat, this is easy; afterwards to find it again and get it out, this 'is practically impossible. All this Maxwell and Boltzmann have ex- plained; the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his Elementary Principles of Statistical Mechanics. For those who take this point of view, the prin- ciple of Carnot is only an imperfect principle, a sort of concession to the infirmity of our senses; it is because our eyes are too gross that we do not distinguish the elements of the blend; it is because our hands are too gross that we cannot force them to separate; the imaginary demon of Maxwell, who is able to sort the molecules one by one, could well constrain the world to return backward. Can it re- turn of itself? That is not impossible; that is only infinitely improbable. The chances are that we should long await the concourse of circumstances which would permit a retrogradation, but soon or late, they would be realized, after years whose number it would take millions of figures to write. These reservations, however, all remained theo- retic and were not very disquieting, and the prin- ciple of Carnot retained all its practical value. But here the scene changes. The biologist, armed with his microscope, long ago noticed in his preparations disorderly move- rrments of little particles in suspension; this is the Brownian movement; he first thought this was a vital phenomenon, but soon he saw that the in- animate bodies danced with no less ardor than the others; then he turned the matter over to the physicists. Unhappily, the physicists remained long uninterested in this question; one concentrates the light to illuminate the microscopic preparation, thought they; with light goes heat; thence inequal- ities of temperature and in the liquid interior cur- rents wh