, we shall find that the course the system actually follows, of its own accord, is always such that along it the action is a minimum (or a maximum).
Now the principle of action issues, as we have said, from the laws of classical mechanics. Were these laws different, the principle of action would cease to apply, at least in its classical form. A priori, we have no means of deciding whether the laws governing physical phenomena of a non-mechanical nature—those of electromagnetics, for example—would also issue in the same principle of action.
But when Maxwell had proved that his equations of electromagnetics could be thrown into a form compatible with the principle of action, and when he had succeeded in amalgamating electricity, magnetism and optics into one science, the universal validity of the principle was accepted. Inasmuch as this principle includes that of the conservation of energy, we can understand why the principle of action was often referred to as the supreme principle of physical science. Incidentally, we may mention that when the principle of action is satisfied by a phenomenon, an indefinite number of different mechanical interpretations of the phenomenon are theoretically possible. In the case of electrodynamic phenomena, however, in view of the complicated hypotheses which he was compelled to postulate, Maxwell abandoned all attempts to discover the precise mechanical interpretation which would correspond to reality.
Thus far, we have limited ourselves to stating the general validity of the principle of action, as applying to all physical processes. It now remains to be seen how the principle will be of use to us in furthering our knowledge of the laws of nature. This we may understand as follows: The principle, as we have seen, imposes the condition that the natural evolution of any system must be such as to render the action a maximum or a minimum. Could we but express this condition in terms of the usual physical magnitudes, we should be enabled to map out in advance the series of intermediary states through which the phenomenon would pass. From this knowledge we should derive the expression of the laws which governed the evolution of the phenomenon. Here, of course, a twofold problem presents itself. First, we must succeed in finding the correct mathematical expression for the action; and, secondly, we must be in a position to solve the purely mathematical problem of determining under what conditions the action will be a maximum or a minimum.
Now all problems of maxima and minima are solved by means of the calculus of variations, a form of calculus we owe chiefly to Lagrange. According to the methods of this calculus, we establish under what conditions a magnitude is a maximum or a minimum by discovering under what conditions it will be stationary.
Let us explain what is meant by the word “stationary.” When a stone is thrown into the air, it ascends with decreasing speed, then seems to hesitate for a brief period of time as it hovers near the point of maximum height before it starts to fall back again towards the earth. During this brief period of hesitation at the apex of its trajectory, the stone is said to remain “stationary.” We can recognise a stationary state by observing that when it is reached no perceptible changes take place over a short period of time. In this way, we may understand the connection which exists between the stationary condition and the presence of a maximum or a minimum. In mathematics small variations are represented by the letter
; hence the stationary condition of the action, or again, the principle of action, is expressed by
