b. Dublin, 1805; d. Dublin, 1865
Calculus of variations; optics; dynamics; algebra; geometry
Hamilton was educated at Trinity College, Dublin. Shortly before graduating in 1827, he was appointed Professor of Astronomy and Royal Astronomer of Ireland, a post which he held for the remainder of his life.
His first paper, “Theory of systems of rays: part first” (1828), was written while he was still an undergraduate. In it he developed new methods for the study of foci and caustics produced by the reflection of light from curved surfaces. Hamilton developed his approach to optics over the following five years, publishing three substantial supplements to his original paper. He showed that the properties of an optical system are completely determined by a certain “characteristic function” that is a function of the initial and final coordinates of a ray of light and which measures the time of passage of light through the system. In 1832 he predicted that light falling at a certain angle on a biaxial crystal would be refracted to form a hollow cone of emergent rays. This prediction was verified by his friend and colleague Humphrey Lloyd.
Hamilton adapted his optical methods to the study of dynamics. In a paper “On a general method in dynamics” (1834), he showed that the dynamics of a system of attracting and repelling point particles is completely determined by a certain characteristic function, which satisfies a differential equation, today referred to as the HAMILTON-JACOBI EQUATION [IV.12 §2.1]. In a subsequent paper, “Second essay on a general method in dynamics” (1835), he introduced the principal function of a dynamical system, presented the equations of motion of such a system in HAMILTONIAN FORM [IV.16 §2.1.3], and adapted methods of perturbation theory to this setting.
Hamilton discovered the system of QUATERNIONS [III.76] in 1843. The fundamental equations of this system occurred to him in a flash of insight as he was walking along the bank of the Royal Canal, near Dublin, on October 16 of that year. Most of his subsequent mathematical work involved quaternions. It is not difficult to translate much of this work into the language of modern vector analysis, and indeed many of the basic concepts and results of vector algebra and analysis emerged from Hamilton’s work on quaternions. Hamilton applied quaternion methods to the study of dynamics in a series of short papers published in the three years immediately following his discovery of quaternions. He also investigated a number of algebraic systems related to quaternions. However, most of his work with quaternions was concerned with their application to the study of geometrical problems, and, in particular, to the study of surfaces of the second order and (especially in the final years of his life) to the study of the differential geometry of curves and surfaces. Much of this research is to be found in his two books Lectures on Quaternions (1853) and Elements of Quaternions (1866, published posthumously).
Hankins, T. L. 1980. Sir William Rowan Hamilton. Baltimore, MD: Johns Hopkins University Press.