Optics and Electricity
Project Gutenberg EPUB3 edition
Poincaré, Henri and Halsted, George Bruce
That granted, the system will always go from one position to another by a path such that the average action shall be a minimum.
It matters little that T and U are now expressed by the aid of the parameters q and their derivatives; it matters little that it is also by means of these parameters that we define the initial and final positions; the principle of least action remains always true.
Now here again, of all the paths that lead from one position to another, there is one for which the average action is a minimum, and there is only one. The principle of least action suffices, then, to determine the differential equations which define the variations of the parameters q.
The equations thus obtained are another form of the equations of Lagrange.
To form these equations we need to know neither the relations that connect the parameters q with the coordinates of the hypothetical molecules, nor the masses of these molecules, nor the expression of U as a function of the coordinates of these molecules.
All we need to know is the expression of U as a function of the parameters, and that of T as a function of the parameters q and their derivatives, that is, the expressions of the kinetic and of the potential energy as functions of the experimental data.
Then we shall have one of two things: either for a suitable choice of the functions T and U, the equations of Lagrange, constructed as we have just said, will be identical with the differential equations deduced from experiments; or else there will exist no functions T and U, for which this agreement takes place. In the latter case it is clear that no mechanical explanation is possible.
The necessary condition for a mechanical explanation to be possible is therefore that we can choose the functions T and U in such a way as to satisfy the principle of least action, which involves that of the conservation of energy.
This condition, moreover, is sufficient. Suppose, in fact, that we have found a function U of the parameters q, which represents one of the parts of the energy; that another part of the energy, which we shall represent by T, is a function of the parameters q and their derivatives, and that it is a homogeneous polynomial of the second degree with respect to these derivatives; and finally that the equations of Lagrange, formed by means of these two functions, T and U, conform to the data of the experiment.
What is necessary in order to deduce from this a mechanical explanation? It is necessary that U can be regarded as the potential energy of a system and T as the vis viva of the same system.
There is no difficulty as to U, but can T be regarded as the vis viva of a material system?
It is easy to show that this is always possible, and even in an infinity of ways. I will confine myself to referring for more details to the preface of my work, 'Électricité et optique.'
Thus if the principle of least action can not be satisfied, no mechanical explanation is possible; if it can be satisfied, there is not only one, but an infinity, whence it follows that as soon as there is one there is an infinity of others.
One more observation.
Among the quantities that experiment gives us directly, we shall regard some as functions of the coordinates of our hypothetical molecules; these are our parameters q. We shall look upon the others as dependent not only on the coordinates, but on the velocities, or, what comes to the same thing, on the derivatives of the parameters q, or as combinations of these parameters and their derivatives.
And then a question presents itself: among all these quantities measured experimentally, which shall we choose to represent the parameters q? Which shall we prefer to regard as the derivatives of these parameters? This choice remains arbitrary to a very large extent; but, for a mechanical explanation to be possible, it suffices if we can make the choice in such a way as to accord with the principle of least action.
And then Maxwell asked himself whether he could make this choice and that of the two energies T and U, in such a way that the electrical phenomena would satisfy this principle. Experiment shows us that the energy of an electromagnetic field is decomposed into two parts, the electrostatic energy and the electrodynamic energy. Maxwell observed that if we regard the first as representing the potential energy U, the second as representing the kinetic energy T; if, moreover, the electrostatic charges of the conductors are considered as parameters q and the intensities of the currents as the derivatives of other parameters q; under these conditions, I say, Maxwell observed that the electric phenomena satisfy the principle of least action. Thenceforth he was certain of the possibility of a mechanical explanation.
If he had explained this idea at the beginning of his book instead of relegating it to an obscure part of the second volume, it would not have escaped the majority of readers.
If, then, a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others, that will render an account equally well of all the particulars revealed by experiment.
And this is confirmed by the history of every branch of physics; in optics, for instance, Fresnel believed vibration to be perpendicular to the plane of polarization; Neumann regarded it as parallel to this plane. An 'experimentum crucis' has long been sought which would enable us to decide between these two theories, but it has not been found.
In the same way, without leaving the domain of electricity, we may ascertain that the theory of two fluids and that of the single fluid both account in a fashion equally satisfactory for all the observed laws of electrostatics.
All these facts are easily explicable, thanks to the properties of the equations of Lagrange which I have just recalled.
It is easy now to comprehend what is Maxwell's fundamental idea.
To demonstrate the possibility of a mechanical explanation of electricity, we need not preoccupy ourselves with finding this explanation itself; it suffices us to know the expression of the two functions T and U, which are the two parts of energy, to form with these two functions the equations of Lagrange and then to compare these equations with the experimental laws.
Among all these possible explanations, how make a choice for which the aid of experiment fails us? A day will come perhaps when physicists will not interest themselves in these questions, inaccessible to positive methods, and will abandon them to the metaphysicians. This day has not yet arrived; man does not resign himself so easily to be forever ignorant of the foundation of things.
Our choice can therefore be further guided only by considerations where the part of personal appreciation is very great; there are, however, solutions that all the world will reject because of their whimsicality, and others that all the world will prefer because of their simplicity.
In what concerns electricity and magnetism, Maxwell abstains from making any choice. It is not that he systematically disdains all that is unattainable by positive methods; the time he has devoted to the kinetic theory of gases sufficiently proves that. I will add that if, in his great work, he develops no complete explanation, he had previously attempted to give one in an article in the Philosophical Magazine. The strangeness and the complexity of the hypotheses he had been obliged to make had led him afterwards to give this up.
The same spirit is found throughout the whole work. What is essential, that is to say what must remain common to all theories, is made prominent; all that would only be suitable to a particular theory is nearly always passed over in silence. Thus the reader finds himself in the presence of a form almost devoid of matter, which he is at first tempted to take for a fugitive shadow not to be grasped. But the efforts to which he is thus condemned force him to think and he ends by comprehending what was often rather artificial in the theoretic constructs he had previously only wondered at.