Energy And Thermo-dynamics

Energetics

The difficulties raised by the classical mechanics have led certain minds to prefer a new system called “Energetics”.

Energetics came from the discovery of the principle of the conservation of energy.

• Helmholtz gave it its definite form.

Its fundamental parts are kinetic and potential energy.

Every change that the bodies of nature can undergo is regulated by two experimental laws.

First, the sum of the kinetic and potential energies is constant. This is the principle of the conservation of energy.

Second, if a system of bodies is at A at the time t 0 , and at B at the time t 1 , it always passes from the first position to the second by such a path that the mean value of the difference between the 2 kinds of energy in the interval of time which separates the 2 epochs t 0 and t 1 is a minimum.

This is Hamilton’s principle, and is one of the forms of the principle of least action.

The energetic theory has the following advantages over the classical. First, it is less incomplete—that is to say, the principles of the conservation of energy and of Hamilton teach us more than the fundamental principles of the classical theory, and exclude certain motions which do not occur in nature and which would be compatible with the classical theory.

Second, it frees us from the hypothesis of atoms, which it was almost impossible to avoid with the classical theory. But in its turn it raises fresh difficulties.

The definitions of the two kinds of energy would raise difficulties almost as great as those of force and mass in the first system.

However, we can get out of these difficulties more easily, at any rate in the simplest cases.

Assume an isolated system formed of a certain number of material points. Assume that these points are acted upon by forces depending only on their relative position and their distances apart, and indepen- dent of their velocities. In virtue of the principle of the conservation of energy there must be a function of forces.

In this simple case the enunciation of the principle of the conservation of energy is of extreme simplicity. A certain quantity, which may be determined by experiment, must remain constant. This quantity is the sum of 2 terms.

The first depends only on the position of the material points, and is independent of their velocities; the second is proportional to the squares of these velocities.

This decomposition can only take place in one way.

The first of these is potential energy. The second is kinetic energy.

If T + U is 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, and the other proportional to the square of the velocities. Among the functions which remain constant there is only one which enjoys this property.

It is T + U (or a linear function of T + U, it matters not which, since this linear function may always be reduced to T + U by a change of unit and of origin). This, then, is what we call energy.

The first term we shall call potential energy, and the second kinetic energy. The definition of the two kinds of energy may therefore be carried through without any ambiguity.

So it is with the definition of mass. Kinetic energy, or vis viva, is expressed very simply by the aid of the masses, and of the relative velocities of all the material points with reference to one of them. These relative velocities may be observed, and when we have the expression of the kinetic energy as a function of these relative velocities, the co-efficients of this expression will give us the masses.

So in this simple case the fundamental ideas can be defined without difficulty. But the difficulties reappear in the more complicated cases if the forces, instead of depending solely 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 on their acceleration.

If material points attracted each other according to an analogous law, U would depend on the velocity, and it might contain a term proportional to the square of the velocity. How can we detect among such terms those that arise from T or U? and how, therefore, can we distinguish the two parts of the energy? But there is more than this. How can we define energy itself? We have no more reason to take as our definition T + U rather than any other function of T+U, when the property which characterised T + U has disappeared—namely, that of being the sum of two terms of a particular form. But that is not all.

We must take account, not only of mechanical energy prop- erly so called, but of the other forms of energy—heat, chemical energy, electrical energy, etc. The principle of the conservation of energy must be written T+U+Q = a constant, where T is the sensible kinetic energy, U the po- tential energy of position, depending only on the position of the bodies, Q the internal molecular energy under the thermal, chemical, or electrical form. This would be all right if the 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 depending only on their internal state. The expression for the energy could be decomposed in one way only into three terms of this form. But this is not the case. Let us consider electrified bodies.

The electrostatic energy due to their mutual action will evidently depend on their charge—i.e., on their state; but it will equally depend on their position. If these bodies are in motion, they will act electro-dynamically on one another, and the electro-dynamic energy will depend not only on their state and their position but on their velocities. We have therefore no means of making the selection of the terms which should form part of T, and U, and Q, and of separating the three parts of the energy. If T + U + Q is constant, the same is true of any function whatever, φ(T + U + Q).

If T + U + Q were of the particular form that I have suggested above, no ambiguity would ensue. Among the functions φ(T + U + Q) which remain constant, there is only one that would be of this particular form, namely the one which I would agree to call energy. But I have saidenergy and thermo-dynamics.

this is not rigorously the case. Among the functions that remain constant there is not one which can rigorously be placed in this particular form. How then can we choose from among them that which should be called energy? We have no longer any guide in our choice. Of the principle of the conservation of energy there is nothing left then but an enunciation:—There is some- thing which remains constant. In this form it, in its turn, is outside the bounds of experiment and reduced to a kind of tautology. It is clear that if the world is governed by laws there will be quantities which remain constant. Like Newton’s laws, and for an analogous reason, the principle of the conservation of energy being based on experiment, can no longer be invalidated by it. This discussion shows that, in passing from the clas- sical system to the energetic, an advance has been made; but it shows, at the same time, that we have not advanced far enough. Another objection seems to be still more serious. The principle of least action is applicable to reversible phe- nomena, but it is by no means satisfactory as far as irreversible phenomena are concerned.

Helmholtz attempted to extend it to this class of phenomena, but he did not and could not succeed. So far as this is concerned all has yet to be done.

The very enunciation of the principle of least action is objectionable. To move from one point to another, a material molecule, acted upon by no force, but compelled to move on a surface, will take as its path the geodesic line—i.e., the shortest path.

This molecule seems to know the point to which we want to take it, to foresee the time that it will take it to reach it by such a path, and then to know how to choose the most convenient path.

The enunciation of the principle presents it to us as a living and free entity.

It would be better to replace it by a less objectionable enunciation, one in which, as philosophers would say, final effects do not seem to be substituted for acting causes.