# Hamilton, Sir William Rowan

### physics action algebra light

(1805–65) Irish mathematician: invented quaternions and a new theory of dynamics.

Hamilton was born in Dublin. His father, a solicitor, sent him to Trim to be raised by his aunt and an eccentric clergyman-linguist uncle when he was 3. He showed an early and astonishing ability at languages (he had mastered 13 by age 13) and later also wrote rather bad poetry and corresponded with Wordsworth, Coleridge and Southey. At 10 he developed an interest in the mathematical classics, including those by , and later went to Trinity College, Dublin. There he did original research on caustics (patterns produced by reflected light).

At Trinity College Hamilton was one of the very few people to have obtained the highest grade ( *optime* ) in two subjects: Greek and mathematical physics. His work on caustics led to his discovery of the law of least action: for a light path the action is a simple function of its length and the light travels along a line minimizing this. Such least action principles dependent on a function of the path taken can powerfully express many laws of physics previously given in more clumsy differential equation form.

At age 22, and before he graduated, Hamilton was appointed professor of astronomy at Dublin (1827) and made Astronomer Royal of Ireland, so that he would be free to do research. He was not a good practical astronomer, despite engaging three of his many sisters to live at the Dunsink observatory to help him.

The field of complex numbers interested him, and he invented quaternions. From 1833 he had considered a + *i* b as an ordered pair (a,b) and considered how rotations in a plane were described by the algebra of such couples. The quaternion refers to a triple and describes rotations in three dimensions. This was an algebra in which (for the first time) the commutative principle that *ij* = *ji* broke down. For a quaternion a + b *i* + c *j* + d *k* , *i* 2 = *j* 2 = *k* 2 = *ijk* = –1 and *ij* = – *ji* . While important for the way that the concepts of algebra were generalized, the subject never had major uses in physics, which was better served by vector and tensor analysis.

In his later years Hamilton became a recluse, working and drinking excessively. His name remains familiar in the Hamiltonian operators of quantum mechanics.

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