What is it necessary to do to give a mechanical interpretation of such a phenomenon?
One will try to explain it either by the motions of ordinary matter, or by those of one or more hypothetical fluids.
These fluids will be considered as formed of a very great number of isolated molecules m.
When shall we say, then, that we have a complete mechanical explanation of the phenomenon? It will be, on the one hand, when we know the differential equations satisfied by the coordinates of these hypothetical molecules m, equations which, moreover, must conform to the principles of dynamics; and, on the other hand, when we know the relations that define the coordinates of the molecules m as functions of the parameters q accessible to experiment.
These equations, as I have said, must conform to the principles of dynamics, and, in particular, to the principle of the conservation of energy and the principle of least action.
The first of these two principles teaches us that the total energy is constant and that this energy is divided into two parts:
1º The kinetic energy, or vis viva, which depends on the masses of the hypothetical molecules m, and their velocities, and which I shall call T.
2º The potential energy, which depends only on the coordinates of these molecules and which I shall call U. It is the sum of the two energies T and U which is constant.
What now does the principle of least action tell us? It tells us that to pass from the initial position occupied at the instant t0 to the final position occupied at the instant t1, the system must take such a path that, in the interval of time that elapses between the two instants t0 and t1, the average value of 'the action' (that is to say, of the difference between the two energies T and U) shall be as small as possible.
If the two functions T and U are known, this principle suffices to determine the equations of motion.