William Rowan Hamilton

Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathematical equations are considered fundamental to modern theoretical physics, particularly his reformulation of Lagrangian mechanics. His career included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra.

Hamilton was Andrews Professor of Astronomy at Trinity College Dublin. He was also the third director of Dunsink Observatory from 1827 to 1865. The Hamilton Institute at Maynooth University is named after him.

Early life

Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street, later renumbered to 36. Hamilton's father, who was from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, County Meath.

Hamilton is said to have shown talent at an early age. His uncle observed that Hamilton, from a young age, had displayed an uncanny ability to acquire languages — a claim which has been disputed by some historians, who claim he had only a basic understanding of them.Bruno (2003) At the age of seven, he had already made progress in Hebrew, and before he was 13, he had acquired, under the care of his uncle, a dozen languages: classical and modern European languages, Persian, Arabic, Hindustani, Sanskrit, Marathi and Malay. The emphasis of Hamilton's early education on languages is attributed to the wish of his father to see him employed by the British East India Company.

An expert mental calculator, the young Hamilton was capable of working out some calculations to many decimal places. In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton. The two were pitted against each other in a mental arithmetic contest, with Colburn emerging as the clear victor.

In reaction to his defeat, Hamilton spent less time studying languages, and more on mathematics. At age ten, he stumbled across a Latin copy of Euclid; and at twelve he studied Newton's '' Arithmetica Universalis''. By age 16, he had covered much of the '' Principia'', as well as some more recent works on analytic geometry and differential calculus.

Student years

In mid-1822, Hamilton began a systematic study of Laplace's '' Mécanique Céleste''. During this period, he encountered what he believed to be a logical error in ''Mécanique Céleste'', an observation which led Hamilton to be introduced to John Brinkley, then Royal Astronomer of Ireland. In November and December of 1822, he completed his first three original mathematical papers. On his first visit to Dunsink Observatory, he showed two of them to Brinkley, who requested that the papers be developed further. Hamilton complied, and early in 1823, Brinkley approved the amended version. In July of 1823, Hamilton earned a place at Trinity College Dublin by examination, at age 17. His tutor there was Charles Boyton, a family friend, who brought to his attention the contemporary mathematics published by the group at the École Polytechnique in Paris. John Brinkley remarked of the precocious Hamilton, "This young man, I do not say ''will be'', but ''is'', the first mathematician of his age."

The college awarded Hamilton two ''optime''s, or off-the-chart grades, in Greek and in physics. He was first in every subject and at every examination. He aimed to win a Trinity College fellowship by competitive examination, but this did not happen. Instead, after Brinkley in 1826 was made Bishop of Cloyne, Hamilton was appointed in 1827 to the vacant posts left by Brinkley's departure, Andrews Professor of Astronomy and Royal Astronomer of Ireland. Despite having his undergraduate career cut short in this way, he earned degrees in both classics and mathematics (BA in 1827, MA in 1837).

Personal life and poetry

In 1824, Hamilton was introduced at Edgeworthstown to the novelist Maria Edgeworth, by the Rev. Richard Butler, the vicar of Trim, County Meath to whom his uncle James Hamilton was curate.Hankins (1980) During the same period, his uncle introduced him to the Disney family at Summerhill House, County Meath. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister.Hankins (1980)

Hamilton was attracted to Catherine Disney, but her family did not approve and Catherine was required to marry the Rev. William Barlow, a brother of her elder sister's husband. The wedding took place in 1825.Hankins (1980) Hamilton wrote in 1826 about his feelings for her in an extended poem, "The Enthusiast". Over twenty years later, in 1847, he confided in John Herschel that during this period he might have become a poet.

In 1825, Hamilton met Arabella Lawrence, younger sister of Sarah Lawrence, a significant correspondent and frank critic of his poetry. It was a contact he made through Maria Edgeworth's circle.

At Dunsink Observatory

Hamilton, now Royal Astronomer of Ireland, took up residence at Dunsink Observatory where he spent the rest of his life. He was there from 1827 until his death in 1865. In his early years at Dunsink, he observed the heavens quite regularly; He left routine observation to his assistant Charles Thompson. His sisters also supported the observatory's work.

The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women. Felicia Hemans wrote her poem ''The Prayer of the Lonely Student'' after hearing one of his lectures.

Personal life, travel and poetic visits

Hamilton invited his four sisters to come and live at the observatory in 1827, and they ran the household until his marriage in 1833. They included Eliza Mary Hamilton (1807–1851), the poet. In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin.

Newly appointed to the Observatory, Hamilton set off on a tour in Ireland and England with Alexander Nimmo, who was coaching him on latitude and longitude. One call was to Sarah Lawrence's school at Gateacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes. They visited William Wordsworth at Rydal Mount in September of that year, where the writer Caesar Otway was also present.Barker (2001) After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple".

When Wordsworth visited Dublin in the summer of 1829, in a party with John Marshall and his family, he stayed at Dunsink with Hamilton. On a second tour in England with Nimmo in 1831, Hamilton parted from him at Birmingham, to visit the Lawrence sisters and family on his mother's side in the Liverpool area. They met up again in the Lake District, where they climbed Helvellyn and had tea with Wordsworth. Hamilton returned to Dublin, via Edinburgh and Glasgow.

Hamilton visited Samuel Taylor Coleridge at Highgate, in 1832, helped by an unexpected letter of introduction given to him by Sarah Lawrence on a visit to Liverpool in March of that year. He also paid a call, with Arabella, on the family of William Roscoe, who had died in 1831.

He was a Christian, described as "a lover of the Bible, an orthodox and attached member of the Established Church", and as having a "profound conviction of the truth of revealed religion".

Family

While attending Trinity College, Hamilton proposed to his friend's sister, whose refusal drove the young Hamilton to depression and illness, even to the verge of suicide. He proposed again in 1831 to Ellen de Vere, a sister of the poet Aubrey De Vere, who declined as well. Hamilton eventually married Helen Marie Bayly in 1833, a country preacher's daughter, and had three children with her: William Edwin Hamilton (born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840). Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, timid, and chronically ill.

Death

Hamilton retained his faculties unimpaired to the last, and continued the task of finishing the ''Elements of Quaternions'' which had occupied the last six years of his life. He died on 2 September 1865, following a severe attack of gout. He is buried in Mount Jerome Cemetery in Dublin.

Physics

Hamilton made outstanding contributions to classical mechanics and optics.

His first discovery was in an early paper that he communicated in 1823 to John Brinkley, who presented it under the title of ''Caustics'' in 1824 to the Royal Irish Academy. It was referred as usual to a committee, which recommended further development and simplification before publication. Between 1825 and 1828 the paper was expanded, and became a clearer exposition of a novel method. Over this period, Hamilton gained an appreciation for the nature and importance of optics.

In 1827, Hamilton presented a theory of a single function, now known as Hamilton's principal function, that brings together mechanics and optical theory. It helped to establish the foundations of the wave theory of light in mathematical physics. He proposed it when he first predicted its existence in the third supplement to his ''Systems of Rays'', read in 1832.

The Royal Irish Academy paper was finally entitled ''Theory of Systems of Rays'' (23 April 1827), and the first part was printed in 1828 in the ''Transactions of the Royal Irish Academy''. The more important contents of the second and third parts appeared in the three voluminous supplements (to the first part) which were published in the same Transactions, and in the two papers ''On a General Method in Dynamics'', which appeared in the ''Philosophical Transactions'' in 1834 and 1835. In these papers, Hamilton developed his central principle of "Varying Action".

A result of this work is a prediction for transparent biaxial crystals (i.e. monoclinic, orthorhombic or triclinic crystals). A ray of light entering such a crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known as conical refraction. Hamilton found it from the geometry of the wave surface introduced by Augustin-Jean Fresnel, which has singular point. There is a basic mathematical explanation of the phenomenon, namely that the wave surface is not the boundary of a convex body. A fuller understanding awaited the microlocal analysis of the middle of the 20th century,

The step from optics to dynamics in the application of the method of "Varying Action" was made in 1827, and communicated to the Royal Society, in whose ''Philosophical Transactions'' for 1834 and 1835 there are two papers on the subject.

Context and importance of the work

Hamiltonian mechanics was a powerful new technique for working with equations of motion. Hamilton's advances enlarged the class of mechanical problems that could be solved. His principle of "Varying Action" was based on the calculus of variations, in the general class of problems included under the principle of least action which had been studied earlier by Pierre Louis Maupertuis, Euler, Joseph Louis Lagrange

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