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