. Henceforth (
) or
will be referred to as the function of action.[123]
Roughly speaking, action was thus in the nature of the product of a duration by an energy contained in a volume of space. On no account may this action be confused with the action dealt with in Newton’s law of action and reaction, also expressible as the principle of the conservation of momentum. Still less may it be confused with the term “action” which appears in philosophical writings. The importance of the conception of action arises from the fact that the laws of mechanics can be expressed in a highly condensed form when the conception of action is taken into account. We thus obtain what is known as the Principle of Least Action, with which the names of Fermat, Maupertius and Lagrange are associated. A more convenient form of the principle was given by Hamilton; it then becomes Hamilton’s Principle of Stationary Action. If we restrict our attention to the very simplest case, we may state Hamilton’s principle as follows:
If we consider all the varied paths along which a conservative system may be guided, so that it will pass in a given time from a definite initial configuration
to a definite configuration