The calculus of variations enables us to solve such purely mathematical problems. Under the circumstances the problem of determining the laws which govern any particular phenomenon reduces to discovering the expression of the function of action
pertaining to the phenomenon in question. Larmor applied this method to the phenomena of electricity and magnetism and showed how Maxwell’s laws of electrodynamics could be deduced from a suitable mathematical expression
defining the electromagnetic function of action.
When the theory of relativity supplanted classical science, it was recognised that the classical equations of mechanics were only approximate, and it became necessary to reformulate the principle of action so as to render it compatible with the mechanics of relativity, hence also with space-time. This work was carried out by the pure mathematicians—by Klein and Hilbert in particular. It was then found that a principle of action differing but slightly from the classical one could be obtained. But inasmuch as the difference involved presents only a theoretical interest, we shall discuss the bearing of the principle of action on the theory of relativity without taking the slight aforementioned modification into consideration.
In classical science, it was strange to find that action, though so important in its physical significance, should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time. Designating a volume of space-time by
, we have
