W. R. Hamilton's Characterisitic Function and the "Principle of Least Action " in His Research on Optics

Abstract

W. R. Hamilton proposed, in his articles written between 1824 and 1833, the characteristic function to describe the properties of ray systems. It has been usually presumed that he derived the function from the principle of least action. The present paper aims at revealing the historical connection between the above mentioned function and principle. Firstly discussed are the origin and implication of the principle which he called that of least action. He derived it from the law of reflection and refraction ; what he called ACTION in this process did not imply mechanical action but optical path length. Accordingly, the principle should originally be called that of the shortest optical path length. Secondly dealt with are the basis of his characteristic function and the way of its extension. It was introduced, not on the basis of the above-mentioned principle, but through analysis of rays in focal mirrors and focal refractors; before long, he noticed that the function was compatible with the principle ; then, for the purpose of expressing the ray systems in inhomogeneous medium, he utilized a refined form of the function ; besides, he formulated the function on the basis of the principle; finally the principle was accepted as an axiom.