Sir William Rowan Hamilton | |
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Sir William Rowan Hamilton (1805–1865) | |
| Born | 4 August 1805 Dublin, Ireland |
| Died | 2 September 1865 (aged 60) Dublin, Ireland |
| Nationality | United Kingdom of Great Britian and Ireland |
| Alma mater | Trinity College Dublin |
| Known for | Hamilton's principle Hamiltonian mechanics Hamiltonians Hamilton–Jacobi equation Quaternions Biquaternions Hamiltonian path Icosian calculus Nabla symbol Versor Coining the word 'tensor' Coining the word 'scalar' cis notation Hamiltonian vector field Icosian game Universal algebra Hodograph Hamiltonian group Cayley–Hamilton theorem |
| Spouse(s) | Helen Maria Bayly |
| Children | William Edwin Hamilton, Archibald Henry Hamilton, Helen Eliza Amelia O'Regan, née Hamilton |
| Awards | Royal Medal (1835) |
| Scientific career | |
| Fields | Mathematics, astronomy, physics |
| Institutions | Trinity College, Dublin |
| Academic advisors | John Brinkley |
| Influences | Zerah Colburn John T. Graves |
| Influenced | Peter Guthrie Tait |
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865)[1][2] was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and a director at Dunsink Observatory. He made major contributions to optics, classical mechanics and abstract algebra. His work was of importance to theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions.
Hamilton's scientific career included the study of geometrical optics, adaptation of dynamic methods in optical systems, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley–Hamilton theorem). Hamilton also invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.
Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),[3] who lived in Dublin at 29 Dominick Street, later renumbered to 36.[4] 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,[3] a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. Meath.[5] James's daughter Grace, Hamilton's cousin, became the mother of Mary Elizabeth Townsend, philanthropist and co-founder of the Girls' Friendly Society.[6]
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. This has been disputed by some historians, who claim he had only a basic understanding of them.[3]: 207 At the age of seven, he had already made considerable progress in Hebrew, and before he was thirteen he had acquired, under the care of his uncle (a linguist), almost as many languages as he had years of age. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, and even Marathi and Malay. He retained much of his linguistic knowledge to the end of his life.
An expert mental calculator, the young Hamilton was capable of working out the result of some calculation to many decimal places. In September 1813, an 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 the clear victor.[3]: 208 In reaction to his defeat, Hamilton spent less time on studying languages, and more on mathematics.[7][8][9] Aged ten, he stumbled across a Latin copy of Euclid; and at twelve he studied Newton's Arithmetica Universalis. He moved on to read the Principia, and by age 16 he had covered a much part of it, as well as some more recent works on analytic geometry and the differential calculus.[10]
Hamilton was also preparing to enter Trinity College, Dublin, and therefore had to devote some time to classics. Hamilton's predecessor as Royal Astronomer of Ireland, John Brinkley, remarked of the 18-year-old Hamilton, "This young man, I do not say will be, but is, the first mathematician of his age."[11]
Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College in Dublin, which he entered at age 18. In mid-1822 he began a systematic study of Laplace's Mécanique Céleste.
The college awarded him two optimes, or off-the-chart grades. He was in fact first in every subject and at every examination. He achieved the rare distinction of obtaining an optime both for Greek and for physics. He was expected to win more student honours, but his undergraduate career was curtailed.[7] He did take degrees in both classics and mathematics (BA in 1827, MA in 1837); but remarkably, still an undergraduate, he was appointed Andrews Professor of Astronomy and Royal Astronomer of Ireland in 1827.[3]: 209 He then took up residence at Dunsink Observatory where he spent the rest of his life.[8]
As a student Hamilton had been attracted to Catherine Disney, sister of one of his Trinity College friends. Her family did not approve, and Catherine was forced to marry the reverend William Barlow, a brother of her elder sister's husband. The wedding took place in 1825.[12]: 109, 113 He had met Catherine in August 1824, in the family circle of Maria Edgeworth. He 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 at this period he might have become a poet.[13]
In 1825 Hamilton met Arabella Lawrence, younger sister of Sarah Lawrence, a significant correspondent and frank critic of his poetry. It was a contact made also through Maria Edgeworth's circle.[12]: 26 [14]
On 16 June 1827, just 21 years old and still an undergraduate, Hamilton was elected Royal Astronomer of Ireland and went to live at Dunsink Observatory where he remained until his death in 1865.[15] In his youth he had owned a telescope;[16] he calculated celestial phenomena, for instance the locations of the visibility of eclipses of the moon.[17]
In his early years at Dunsink, Hamilton observed the heavens quite regularly.[18] Ultimately he left routine observation to his assistant Charles Thompson.[19][20] His introductory lectures in astronomy were famous; in addition to his students, they attracted scholars and poets, and women;[21] Felicia Hemans wrote her poem The Prayer of the Lonely Student after hearing one of his lectures.[22]
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.[5] In 1827 Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin.[23][24]
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.[25] One call was to Sarah Lawrence's school at Gatacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes.[26] They visited William Wordsworth at Rydal Mount in September of that year, where Caesar Otway was also present.[27][28]: 410 After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple".[29]
When Wordsworth visited Dublin in summer 1829, in a party with John Marshall and his family, he stayed at Dunsink with Hamilton.[30]: 411 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.[12][31]
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.[32][33]
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.[7][34] He is buried in Mount Jerome Cemetery in Dublin.
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| Classical mechanics |
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Hamilton made important contributions to optics and to classical mechanics.
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.[10] Over this period Hamilton gained appreciation of the nature and importance of optics.[35]
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 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 great principle of "Varying Action". The most remarkable result of this work is the prediction that a single ray of light entering a biaxial crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known as "conical refraction".[10]
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, which, like the Systems of Rays, display a mastery over symbols and a flow of mathematical language almost unequaled.
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 and others. Hamilton's analysis uncovered deeper mathematical structure than had been previously understood, in particular a symmetry between momentum and position. The credit for discovering what are now called the Lagrangian and Lagrange's equations belongs also to Hamilton.
Both the Lagrangian mechanics and Hamiltonian approaches have proven important in the study of continuous classical systems in physics, and quantum mechanical systems: the techniques find use in electromagnetism, quantum mechanics, relativity theory and quantum field theory. In the Dictionary of Irish Biography David Spearman writes:[36]
The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated. The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this the most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting-point for calculation in almost any area of physics.
Many scientists, including Liouville, Jacobi, Darboux, Poincaré, Kolmogorov, Prigogine[37] and Arnold, have extended Hamilton's work, in mechanics, differential equations and symplectic geometry.[38]
Hamilton's mathematical studies seem to have been undertaken and carried to their full development without any assistance whatsoever, and the result is that his writings do not belong to any particular "school".
Hamilton appears to have devoted himself almost wholly to mathematics, though he always kept himself well acquainted with the progress of science both in Britain and abroad. Hamilton found an important defect in one of Laplace's demonstrations, and he was induced by a friend to write out his remarks, so that they could be shown to Dr. John Brinkley, then the first Royal Astronomer of Ireland, and an accomplished mathematician. Brinkley seems to have immediately perceived Hamilton's talents, and to have encouraged him in the kindest way.
Hamilton's was appointed to the Andrews Professor of Astronomy in the University of Dublin, vacated by John Brinkley in 1827. The chair was not exactly offered to him, as has been sometimes asserted, but the electors, having met and talked over the subject, authorised Hamilton's personal friend (also an elector) to urge Hamilton to become a candidate, a step which Hamilton's modesty had prevented him from taking. Thus, when barely 22, Hamilton was established at the Dunsink Observatory, near Dublin.[10]
Hamilton was not especially suited for the post, because although he had a profound acquaintance with theoretical astronomy, he had paid little attention to the regular work of the practical astronomer. Hamilton's time was better employed in original investigations than it would have been spent in observations made even with the best of instruments. Hamilton was intended by the university authorities who elected him to the professorship of astronomy to spend his time as he best could for the advancement of science, without being tied down to any particular branch. If Hamilton had devoted himself to practical astronomy, the University of Dublin would assuredly have furnished him with instruments and an adequate staff of assistants.[10]
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Main article: History of quaternions |
Hamilton made his discovery of the algebra of quaternions in 1843.[3]: 210 Among much prior related work, in 1840 Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name.[39]
Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional Argand diagram) to higher spatial dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation
occurred to him; Hamilton then carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge).[3]: 210
The quaternions involved abandoning the commutative law, a radical step for the time. In the context of this prototype geometric algebra, Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product as scalar. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the "scalar" part, and the remaining three as the "vector" part. He coined the neologisms "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.[40]
It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in Alice in Wonderland. In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry.[41]
Hamilton looked into the solution of the quintic in the theory of equations, examining of the results arrived at by Niels Henrik Abel, George Jerrard and others in their researches. There is Hamilton's paper on fluctuating functions in Fourier analysis, and the invention of the hodograph. Of his investigations into the solutions, especially by numerical approximation, of certain classes of physically-important differential equations, only parts were published, at intervals, in the Philosophical Magazine.[10]
Hamilton also introduced the icosian game or Hamilton's puzzle which can be solved using the concept of a Hamiltonian path in graph theory.
Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by introduction of complex number coefficients. When his work was assembled in 1853, the book Lectures on Quaternions had "formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in the Halls of Trinity College, Dublin". Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research.
When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the Elements of Quaternions, a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition was prepared by Charles Jasper Joly, when the book was split into two volumes, the first appearing 1899 and the second in 1901. The subject index and footnotes in this second edition improved the Elements accessibility.
Hamilton was twice awarded the Cunningham Medal of the Royal Irish Academy.[42] The first award, in 1834, was for his work on conical refraction, for which he also received the Royal Medal of the Royal Society the following year.[43] He was to win it again in 1848.
In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, Hamilton was knighted by the lord-lieutenant. Other honours rapidly succeeded, among which his election in 1837 to the president's chair in the Royal Irish Academy, and the rare distinction of being made a corresponding member of the Saint Petersburg Academy of Sciences. Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.[44]
A plaque under the Broom Bridge, associated with the discovery of quaternions, was unveiled by Éamon de Valera on 13 November 1958.[45][46] Since 1989, the National University of Ireland, Maynooth has organised a pilgrimage called the Hamilton Walk, in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery.[47]
The Hamilton Institute is an applied mathematics research institute at Maynooth University and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann, Frank Wilczek, Andrew Wiles and Timothy Gowers have all spoken. The year 2005 was the 200th anniversary of Hamilton's birth and the Irish government designated that the Hamilton Year, celebrating Irish science. Trinity College Dublin marked the year by launching the Hamilton Mathematics Institute.[48]
Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions.[49] A 10-euro commemorative silver proof coin was issued by the Central Bank of Ireland in 2005 to commemorate 200 years since his birth.
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Main article: List of things named after William Rowan Hamilton |
Hamilton married Helen Bayly, daughter of Rev Henry Bayly, Rector of Nenagh, County Tipperary, in 1833; she was a sister of neighbours to the observatory.[51][12]: 108 They had three children: William Edwin Hamilton (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840).[52] Helen stayed with her widowed mother at Bayly Farm, Nenagh for extended periods, until her mother's death in 1837. She also was away from Dunsink, staying with sisters, for much of the time from 1840 to 1842.[53] Hamilton's married life was reportedly difficult.[3]: 209 In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression.[12]: 125, 6
