Among all the possible ways of passing from one position to another, there is evidently one for which the average value of the action is less than for any other. There is, moreover, only one; and it results from this that the principle of least action suffices to determine the path followed and consequently the equations of motion.
Thus we obtain what are called the equations of Lagrange.
In these equations, the independent variables are the coordinates of the hypothetical molecules m; but I now suppose that one takes as variables the parameters q directly accessible to experiment.
The two parts of the energy must then be expressed as functions of the parameters q and of their derivatives. They will evidently appear under this form to the experimenter. The latter will naturally try to define the potential and the kinetic energy by the aid of quantities that he can directly observe.[6]
That granted, the system will always go from one position to another by a path such that the average action shall be a minimum.
It matters little that T and U are now expressed by the aid of the parameters q and their derivatives; it matters little that it is also by means of these parameters that we define the initial and final positions; the principle of least action remains always true.
Now here again, of all the paths that lead from one position to another, there is one for which the average action is a minimum, and there is only one. The principle of least action suffices, then, to determine the differential equations which define the variations of the parameters q.
The equations thus obtained are another form of the equations of Lagrange.
To form these equations we need to know neither the relations that connect the parameters q with the coordinates of the hypothetical molecules, nor the masses of these molecules, nor the expression of U as a function of the coordinates of these molecules.
All we need to know is the expression of U as a function of the parameters, and that of T as a function of the parameters q and their derivatives, that is, the expressions of the kinetic and of the potential energy as functions of the experimental data.