The Mechanical Explanation of Physical Phenomena

In every physical phenomenon there is a certain number of parameters which are reached directly by experiment, and which can be measured. I shall call them the parameters q.

Observation next teaches us the laws of the variations of these parameters, and these laws can be generally stated in the form of differential equations which connect together the parameters q and time.

What can be done to give a mechanical interpretation to such a phenomenon? We may endeavour to explains it, either by the movements of ordinary matter, or by those of one or more hypothetical fluids.

These fluids will be considered as formed of a very large number of isolated molecules m. When may we say that we have a complete mechanical explanation of the phenomenon?

It will be, on the one hand, when we know the differential equations which are satisfied by the co-ordinates of these hypothetical molecules m, equations which must, in addition, conform to the laws of dynamics; and, on the other hand, when we know the relations which define the co-ordinates of the molecules m as functions of the parameters q, attainable by experiment.

These equations should conform to the principles of dynamics, and, in particular, to the principle of the conservation of energy, and to that of least action.

The first of these 2 principles teaches us that the total energy is constant, and may be divided into 2 parts:—

1. Kinetic energy, or vis viva, which depends on the masses of the hypothetical molecules m, and on their velocities. This I shall call T.
2. The potential energy which depends only on the co-ordinates of these molecules, and this I shall call U. It is the sum of the energies T and U that is constant.

Now what are we taught by the principle of least action?

It teaches us that to pass from the initial position occupied at the instant t 0 to the final position occupied at the instant t 1 , the system must describe such a path that in the interval of time between the instant t 0 and t 1 , the mean value of the action—i.e., the difference between the two energies T and U, must be as small as possible.

The first of these two principles is, moreover, a consequence of the second. If we know the functions T and U, this second principle is sufficient to determine the equations of motion.

Among the paths which enable us to pass from one position to another, there is clearly one for which the mean value of the action is smaller than for all the others.

In addition, there is only one such path; and it follows from this, that the principle of least action is sufficient to determine the path followed, and therefore the equations ofmotion. We thus obtain what are called the equations of Lagrange.

In these equations the independent variables are the co-ordinates of the hypothetical molecules m; but I now assume that we take for the variables the parameters q, which are directly accessible to experiment.

The two parts of the energy should then be expressed as a function of the parameters q and their derivatives;

It is under this form that they will appear to the experimenter. The latter will naturally endeavour to define kinetic and potential energy by the aid of quantities he can directly observe. 1 If this be granted, the system will always proceed from one position to another by such a path that the mean value of the action is 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 the aid of these parameters that we define the initial and final positions; the principle of least action will always remain true.

Of the whole of the paths which lead from one position to another, there is one and only one for which the mean action is a minimum.

The principle of least action is therefore sufficient for the determination of the differential equations which define the variations of the parameters q.

The equations thus obtained are another form of Lagrange’s equations.

To form these equations we need not know the relations which connect the parameters q with the coordinates of the hypothetical molecules, nor the masses of the molecules, nor the expression of U as a function of the co-ordinates of these molecules.

All we need know is the expression of U as a function of the parameters q, and that of T as a function of the parameters q and their derivatives—i.e., the expressions of the kinetic and potential energy in terms of experimental data.

One of two things must now happen.

Either for a convenient choice of T and U the Lagrangian equations, constructed as we have indicated, will be identical with the differential equations deduced from experiment, or there will be no functions T and U for which this identity 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 this: that we may choose the functions T and U so as to satisfy the principle of least action, and of the conservation of energy. Besides, this condition is sufficient.

Suppose, in fact, that we have found a function U of the parameters q, which represents one of the parts of energy, and that the part of the energy which we represent by T is a function of the parameters q and their derivatives; that it is a polynomial of the second degree with respect to its derivatives, and finally that the Lagrangian equations formed by the aid of these two functions T and U are in conformity with the data of the experiment.

We may add that:

• U will depend only on the q parameters
• T will depend on them and their derivatives with respect to time, and will be a homogeneous polynomial of the second degree with respect to these derivatives.

How can we deduce from this a mechanical explanation?

U must be regarded as the potential energy of a system of which T is the kinetic energy. There is no difficulty as far as U is concerned, but can T be regarded as the vis viva of a material system?

It is easily shown that this is always possible, and in an unlimited number of ways. I will be content with referring the reader to the pages of the preface of my Électricité et Optique for further details.

Thus, if the principle of least action cannot be satisfied, no mechanical explanation is possible; if it can be satisfied, there is not only one explanation, but an unlimited number, whence it follows that since there is one there must be an unlimited number.

One more remark. Among the quantities that may be reached by experiment directly we shall consider some as the co-ordinates of our hypothetical molecules, some will be our parameters q, and the rest will be regarded as dependent not only on the co-ordinates but on the velocities—or what comes to the same thing, we look on them as derivatives of the parameters q, or as combinations of these parameters and their derivatives.

Here then a question occurs: among all these quantities measured experimentally which shall we choose to represent the parameters q? and which shall we prefer to regard as the derivatives of these parameters?

This choice remains arbitrary to a large extent, but a mechanical explanation will be possible if it is done so as to satisfy the principle of least action.

Maxwell asks: Can this choice and that of the 2 energies T and U be made so that electric phenomena will satisfy this principle? Experiment shows us that the energy of an electro-magnetic field decomposes into electro-static and electro-dynamic energy.

Maxwell recognised that if we regard the former as the potential energy U, and the latter as the kinetic energy T, and that if on the other hand we take the electro-static charges of the conductors as the parameters q, and the intensity of the currents as derivatives of other parameters q—under these conditions, Maxwell has recognised that electric phenomena satisfy the principle of least action.

He was then certain of a mechanical explanation. If he had expounded this theory at the beginning of his first volume, instead of relegating it to a corner of the second, it would not have escaped the attention of most readers.

If therefore a phenomenon allows of a complete mechanical explanation, it allows of an unlimited number of others, which will equally take into account all the particulars revealed by experiment.

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 polarisation; Neumann holds that it is parallel to that plane.

For a long time an experimentum crucis was sought for, which would enable us to decide between these two theories, but in vain. In the same way, without going out of the domain of electricity, we find that the theory of two fluids and the single fluid theory equally account in a satisfactory manner for all the laws of electro-statics.

All these facts are easily explained, thanks to the properties of the Lagrange equations.

It is easy now to understand Maxwell’s fundamental idea.

To demonstrate the possibility of a mechanical explanation of electricity we need not trouble to find the explanation itself; we need only know the expression of the two functions T and U, which are the two parts of energy, and to form with these two functions Lagrange’s equations, and then to compare these equations with the experimental laws.

How shall we choose from all the possible explanations one in which the help of experiment will be wanting? The day will perhaps come when physicists will no longer concern themselves with questions which are inaccessible to positive methods, and will leave them to the metaphysicians.

That day has not yet come; man does not so easily resign himself to remaining for ever ignorant of the causes of things.

Our choice cannot be therefore any longer guided by considerations in which personal appreciation plays too large a part. There are, however, solutions which all will reject because of their fantastic nature, and others which all will prefer because of their simplicity. As far as magnetism and electricity are concerned,

Maxwell abstained from making any choice. It is not that he has a systematic contempt for all that positive methods cannot reach, as may be seen from the time he has devoted to the kinetic theory of gases.

If in his magnum opus he develops no complete explanation, he has attempted one in an article in the Philosophical Magazine. The strangeness and the complexity of the hypotheses he found himself compelled to make, led him afterwards to withdraw it.

The same spirit is found throughout his whole work.

He throws into relief the essential—i.e., what is common to all theories; everything that suits only a particular theory is passed over almost in silence. The reader therefore finds himself in the presence of form nearly devoid of matter, which at first he is tempted to take as a fugi- tive and unassailable phantom.

But the efforts he is thus compelled to make force him to think, and eventually he sees that there is often something rather artificial in the theoretical “aggregates” which he once admired.