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Sir William Rowan Hamilton MRIA (3 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland. He worked in both pure mathematics and mathematics for physics. He made important contributions to optics, classical mechanics and algebra. Although Hamilton was not a physicist–he regarded himself as a pure mathematician–his work was of major importance to physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, 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.

Life

Early life

Education

Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College in Dublin, which he entered at age 18. The college awarded him two Optimes, or off-the-chart grades. He studied both classics and mathematics (BA in 1827, MA in 1837). While still an undergraduate he was appointed Andrews professor of Astronomy and Royal Astronomer of Ireland. He then took up residence at Dunsink Observatory where he spent the rest of his life.

Personal life

While attending Trinity College, Hamilton proposed to his friend's sister, who rejected him. Hamilton, being a sensitive young man, became sick and depressed, and almost committed suicide. He was rejected again in 1831 by Ellen de Vere, a sister of the poet Aubrey Thomas de Vere (1814-1902). His proposal to Helen Marie Bayly, a country preacher's daughter, was accepted, and they married in 1833. Hamilton had three children with Bayly: William Edwin Hamilton (born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840). Bayly proved to be pious, shy, timid, and chronically ill, and Hamilton's married life was reportedly difficult.

Death and legacy

Hamilton retained his faculties unimpaired to the last, and steadily 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. Hamilton is recognised as one of Ireland's leading scientists and, as Ireland becomes more aware of its scientific heritage, he is increasingly celebrated. 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. Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions. A 10 Euros commemorative silver Proof coin was issued by the Central Bank of Ireland in 2005 to commemorate 200 years since his birth. The newest maintenance depot for the Dublin LUAS tram system has been named after him. It is located adjacent to the Broombridge stop on the Green Line.

Astronomy

In his youth Hamilton owned a telescope, and became an expert at calculating celestial phenomena, for instance the locations of the visibility of eclipses of the moon. As he had received extremely high grades for both the Classics and Science, it was not too unusual that, on 16 June 1827, only 21 years old and still an undergraduate, he was elected Royal Astronomer of Ireland and came to live at Dunsink Observatory where he remained until his death in 1865. In his early years at Dunsink, Hamilton observed the heavens quite regularly. Observational astronomy in those days mostly consisted of measuring star positions, which was not too interesting for a mathematical mind. But the main reason for ultimately leaving the regular observing completely to his astronomy assistant Charles Thompson was that Hamilton frequently suffered from illnesses after having observed. Nowadays Hamilton is not seen as one of the great astronomers but in his lifetime he was. His Introductory lectures in astronomy were famous; in addition to his students, they attracted many scholars and poets, and even ladies—in those days a remarkable feat. The poet Felicia Hemans wrote her poem ''The Prayer of the Lonely Student'' after hearing one of his lectures.

Physics

Mathematics

Quaternions

The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843. However, in 1840, Benjamin Olinde Rodrigues had already reached a result that amounted to their discovery in all but name. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He failed to find a useful 3-dimensional system (in modern terminology, he failed to find a real, three-dimensional skew-field), but in working with four dimensions he created quaternions. 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 : suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). This event marks the discovery of the quaternion group. A plaque under the bridge was unveiled by the Taoiseach Éamon de Valera, himself a mathematician and student of quaternions, on 13 November 1958. 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.Twenty Years of the Hamilton Walk
by Fiacre Ó Cairbre, Department of Mathematics, National University of Ireland, Maynooth (2005), Irish Math. Soc. Bulletin 65 (2010)
The quaternion involved abandoning commutativity, a radical step for the time. Not only this, but Hamilton also invented the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product. 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. Hamilton coined the words tensor and scalar, and was the first to use the word vector in the modern sense. 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. One of the features of Hamilton's quaternion system was the differential operator del which could be used to express the gradient of a vector field or to express the curl. These operations were applied by Clerk Maxwell to the electrical and magnetic studies of Michael Faraday in Maxwell's Treatise on Electricity and Magnetism (1873). Though the del operator continues to be used, the real quaternions fall short as a representation of spacetime. On the other hand, the biquaternion algebra, in the hands of Arthur W. Conway and Ludwik Silberstein, provided representational tools for Minkowski space and the Lorentz group early in the twentieth century. Today, the quaternions are used in computer graphics, control theory, signal processing, and orbital mechanics, mainly for representing rotations/orientations. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining quaternion transformations is more numerically stable than combining many matrix transformations. In control and modelling applications, quaternions do not have a computational singularity (undefined division by zero) that can occur for quarter-turn rotations (90 degrees) that are achievable by many Air, Sea and Space vehicles. In pure mathematics, quaternions show up significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry. 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.

Other original work

Hamilton originally matured his ideas before putting pen to paper. The discoveries, papers, and treatises previously mentioned might well have formed the whole work of a long and laborious life. But not to speak of his enormous collection of books, full to overflowing with new and original matter, which have been handed over to Trinity College, Dublin, the previous mentioned works barely form the greater portion of what Hamilton has published. Hamilton developed the variational principle, which was reformulated later by Carl Gustav Jacob Jacobi. He also introduced the icosian game or ''Hamilton's puzzle'' which can be solved using the concept of a Hamiltonian path. Hamilton's extraordinary investigations connected with the solution of algebraic equations of the fifth degree, and his examination of the results arrived at by N. H. Abel, G. B. Jerrard, and others in their researches on this subject, form another contribution to science. There is next Hamilton's paper on fluctuating functions, a subject which, since the time of Joseph Fourier, has been of immense and ever increasing value in physical applications of mathematics. There is also the extremely ingenious invention of the hodograph. Of his extensive investigations into the solutions (especially by numerical approximation) of certain classes of physical differential equations, only a few items have been published, at intervals, in the ''Philosophical Magazine''. Besides all this, Hamilton was a voluminous correspondent. Often a single letter of Hamilton's occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of Hamilton's mind never to be satisfied with a general understanding of a question; Hamilton pursued the problem until he knew it in all its details. Hamilton was ever courteous and kind in answering applications for assistance in the study of his works, even when his compliance must have cost him much time. He was excessively precise and hard to please with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little compared with the extent of his investigations.

Commemorations of Hamilton

* Hamilton's equations are a formulation of classical mechanics. * Numerous other concepts and objects in mechanics, such as Hamilton's principle, Hamilton's principal function, the Hamilton–Jacobi equation, Cayley-Hamilton theorem are named after Hamilton. * The Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and, in a different sense, a term from graph theory. * The 'Hamilton Society', a student society at the Royal College of Surgeons in Ireland, was founded in his name in 2004. * The algebra of quaternions is usually denoted by , or in blackboard bold by $\mathbb$, in honour of Hamilton. * The Hamilton Building at Trinity College Dublin is named after him.Hamilton Building TCD
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Publications

* Hamilton, Sir W.R. (1853),
Lectures on Quaternions
' Dublin: Hodges and Smith * Hamilton, Sir W.R., Hamilton, W.E. (ed) (1866),
Elements of Quaternions
' London: Longmans, Green, & Co. * Hamilton, W.R. (1833),
Introductory Lecture on Astronomy
' Dublin University Review and Quarterly Magazine Vol. I, Trinity College Dublin * For Hamilton's mathematical papers see David R. Wilkins,

* List of things named after William Rowan Hamilton

References

Sources

* , 474 pages—Primarily biographical but covers the math and physics Hamilton worked on in sufficient detail to give a flavour of the work. * * * * Chow, Tai L. (2013).
Classical Mechanics: Chaper 5: Hamilton Formulation of Mechanics: Description of Motion in Phase Spaces
'. CRC Press,