Then we shall have one of two things: either for a suitable choice of the functions T and U, the equations of Lagrange, constructed as we have just said, will be identical with the differential equations deduced from experiments; or else there will exist no functions T and U, for which this agreement 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 that we can choose the functions T and U in such a way as to satisfy the principle of least action, which involves that of the conservation of energy.
This condition, moreover, is sufficient. Suppose, in fact, that we have found a function U of the parameters q, which represents one of the parts of the energy; that another part of the energy, which we shall represent by T, is a function of the parameters q and their derivatives, and that it is a homogeneous polynomial of the second degree with respect to these derivatives; and finally that the equations of Lagrange, formed by means of these two functions, T and U, conform to the data of the experiment.
What is necessary in order to deduce from this a mechanical explanation? It is necessary that U can be regarded as the potential energy of a system and T as the vis viva of the same system.
There is no difficulty as to U, but can T be regarded as the vis viva of a material system?
It is easy to show that this is always possible, and even in an infinity of ways. I will confine myself to referring for more details to the preface of my work, 'Électricité et optique.'
Thus if the principle of least action can not be satisfied, no mechanical explanation is possible; if it can be satisfied, there is not only one, but an infinity, whence it follows that as soon as there is one there is an infinity of others.
One more observation.
Among the quantities that experiment gives us directly, we shall regard some as functions of the coordinates of our hypothetical molecules; these are our parameters q. We shall look upon the others as dependent not only on the coordinates, but on the velocities, or, what comes to the same thing, on the derivatives of the parameters q, or as combinations of these parameters and their derivatives.
And then a question presents itself: among all these quantities measured experimentally, which shall we choose to represent the parameters q? Which shall we prefer to regard as the derivatives of these parameters? This choice remains arbitrary to a very large extent; but, for a mechanical explanation to be possible, it suffices if we can make the choice in such a way as to accord with the principle of least action.