The first of these two principles teaches us that the total energy is constant, and may be divided into two parts:
(1) Kinetic energy, or vis viva, which depends on the masses of the hypothetical molecules m, and on their velocities. This I shall call T. (2) The potential energy which depends only on the co-ordinates of these molecules, and this I shall call U. It is the sum of the energies T and U that is constant.
Now what are we taught by the principle of least action? It teaches 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 describe such a path that in the interval of time between the instant t0 and t1, the mean value of the action—i.e., the difference between the two energies T and U, must be as small as possible. The first of these two principles is, moreover, a consequence of the second. If we know the functions T and U, this second principle is sufficient to determine the equations of motion.
Among the paths which enable us to pass from one position to another, there is clearly one for which the mean value of the action is smaller than for all the others. In addition, there is only such path; and it follows from this, that the principle of least action is sufficient to determine the path followed, and therefore the equations of motion. We thus obtain what are called the equations of Lagrange. In these equations the independent
