The values of the distances at any instant depend upon their initial values, upon those of their first derivatives and also upon something else. What is this something else?
If we will not admit that this may be simply one of the second derivatives, we have only the choice of hypotheses. Either it may be supposed, as is ordinarily done, that this something else is the absolute orientation of the universe in space, or the rapidity with which this orientation varies; and this supposition may be correct; it is certainly the most convenient solution for geometry; it is not the most satisfactory for the philosopher, because this orientation does not exist.
Or it may be supposed that this something else is the position or the velocity of some invisible body; this has been done by certain persons who have even called it the body alpha, although we are doomed never to know anything of this body but its name. This is an artifice entirely analogous to that of which I spoke at the end of the paragraph devoted to my reflections on the principle of inertia.
But, after all, the difficulty is artificial. Provided the future indications of our instruments can depend only on the indications they have given us or would have given us formerly, this is all that is necessary. Now as to this we may rest easy.
CHAPTER VIII
Energy and Thermodynamics
Energetics.—The difficulties inherent in the classic mechanics have led certain minds to prefer a new system they call energetics.
Energetics took its rise as an outcome of the discovery of the principle of the conservation of energy. Helmholtz gave it its final form.
It begins by defining two quantities which play the fundamental rôle in this theory. They are kinetic energy, or vis viva, and potential energy.
All the changes which bodies in nature can undergo are regulated by two experimental laws:
1º The sum of kinetic energy and potential energy is constant. This is the principle of the conservation of energy.
2º If a system of bodies is at A at the time t0 and at B at the time t1, it always goes from the first situation to the second in such a way that the mean value of the difference between the two sorts of energy, in the interval of time which separates the two epochs t0 and t1, may be as small as possible.
This is Hamilton's principle, which is one of the forms of the principle of least action.
The energetic theory has the following advantages over the classic theory:
1º It is less incomplete; that is to say, Hamilton's principle and that of the conservation of energy teach us more than the fundamental principles of the classic theory, and exclude certain motions not realized in nature and which would be compatible with the classic theory:
2º It saves us the hypothesis of atoms, which it was almost impossible to avoid with the classic theory.
But it raises in its turn new difficulties:
The definitions of the two sorts of energy would raise difficulties almost as great as those of force and mass in the first system. Yet they may be gotten over more easily, at least in the simplest cases.
Suppose an isolated system formed of a certain number of material points; suppose these points subjected to forces depending only on their relative position and their mutual distances, and independent of their velocities. In virtue of the principle of the conservation of energy, a function of forces must exist.
In this simple case the enunciation of the principle of the conservation of energy is of extreme simplicity. A certain quantity, accessible to experiment, must remain constant. This quantity is the sum of two terms; the first depends only on the position of the material points and is independent of their velocities; the second is proportional to the square of these velocities. This resolution can take place only in a single way.
The first of these terms, which I shall call U, will be the potential energy; the second, which I shall call T, will be the kinetic energy.
It is true that if T + U is a constant, so is any function of T + U,
Φ (T + U).
But this function Φ (T + U) will not be the sum of two terms the one independent of the velocities, the other proportional to the square of these velocities. Among the functions which remain constant there is only one which enjoys this property, that is T + U (or a linear function of T + U, which comes to the same thing, since this linear function may always be reduced to T + U by change of unit and of origin). This then is what we shall call energy; the first term we shall call potential energy and the second kinetic energy. The definition of the two sorts of energy can therefore be carried through without any ambiguity.
It is the same with the definition of the masses. Kinetic energy, or vis viva, is expressed very simply by the aid of the masses and the relative velocities of all the material points with reference to one of them. These relative velocities are accessible to observation, and, when we know the expression of the kinetic energy as function of these relative velocities, the coefficients of this expression will give us the masses.
Thus, in this simple case, the fundamental ideas may be defined without difficulty. But the difficulties reappear in the more complicated cases and, for instance, if the forces, in lieu of depending only on the distances, depend also on the velocities. For example, Weber supposes the mutual action of two electric molecules to depend not only on their distance, but on their velocity and their acceleration. If material points should attract each other according to an analogous law, U would depend on the velocity, and might contain a term proportional to the square of the velocity.
Among the terms proportional to the squares of the velocities, how distinguish those which come from T or from U? Consequently, how distinguish the two parts of energy?
But still more; how define energy itself? We no longer have any reason to take as definition T + U rather than any other function of T + U, when the property which characterized T + U has disappeared, that, namely, of being the sum of two terms of a particular form.
But this is not all; it is necessary to take account, not only of mechanical energy properly so called, but of the other forms of energy, heat, chemical energy, electric energy, etc. The principle of the conservation of energy should be written:
T + U + Q = const.
where T would represent the sensible kinetic energy, U the potential energy of position, depending only on the position of the bodies, Q the internal molecular energy, under the thermal, chemic or electric form.
All would go well if these three terms were absolutely distinct, if T were proportional to the square of the velocities, U independent of these velocities and of the state of the bodies, Q independent of the velocities and of the positions of the bodies and dependent only on their internal state.
The expression for the energy could be resolved only in one single way into three terms of this form.
But this is not the case; consider electrified bodies; the electrostatic energy due to their mutual action will evidently depend upon their charge, that is to say, on their state; but it will equally depend upon their position. If these bodies are in motion, they will act one upon another electrodynamically and the electrodynamic