William Rowan Hamilton

William Hamilton
File:Hamilton.jpg William Rowan Hamilton
Born	August 4, 1805 Dublin, Ireland
Died	September 2, 1865 Dublin, Ireland
Residence	Ireland
Nationality	Irish, of Scottish descent
Field	Mathematician, physicist, and astronomer
Institutions	Trinity College Dublin
Alma mater	Trinity College Dublin
Academic advisor	John Brinkley
Known for	Quaternions and Hamiltonians
Religious stance	Anglican
Note that although Hamilton never had a doctoral advisor, scientific genealogy authorities regard the Reverend John Brinkley as Hamilton's equivalent mentor.

Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. His discovery of quaternions is perhaps his best known investigation. Hamilton's work was later significant in the development of quantum mechanics. Hamilton is said to have showed immense talent at a very early age, prompting astronomer Bishop Dr. John Brinkley to remark in 1823 of Hamilton at the age of eighteen: “This young man, I do not say will be, but is, the first mathematician of his age.”

Biography

Early life

Hamilton was the fourth of nine children of Archibald Hamilton, a solicitor, and Sarrah Hutton. He was born at 36 Dominick Street, Dublin.

When he was a year old, he was placed in the care of an uncle and aunt, James and Sydney Hamilton. Hamilton could read from the bible at age three, and by age four he was able to read some Greek, Latin and Hebrew. At six he was attempting translations of Homer and Virgil. Between the ages of nine and ten he picked up sanscrit, Arabic and Persian, while mastering Italian and French. Two years later, he wrote a syriac grammar for publication. But though to the very end of his life he retained much of the singular learning of his childhood and youth, often reading Persian and Arabic in the intervals of sterner pursuits, he had long abandoned them as a study, and employed them merely as a relaxation.

Around this time, Hamilton encountered math prodigy Zerah Colburn, who could do elaborate calculations in his head. Hamilton competed with Colburn, but was never able to match his acuity in the sphere of calculation, although it opened the budding mathematician's eyes to new possibilities.

When Hamilton was 12, he lost his mother, and two years later, his father. When he was 15, he began to tackle science and mathematics, beginning with a study of Isaac Newton's Principia. Around the age of 17, he was tackling the infinitesimal calculus. About this period Hamilton was also engaged in preparation for entrance at Trinity College, Dublin, and had therefore to devote a portion of time to classics. In the summer of 1822, in his seventeenth year, he began a systematic study of Laplace's Mécanique Céleste.

It was in the successful effort to open this treasure-house that Hamilton’s mind received its final temper. From that time Hamilton appears to have devoted himself almost wholly to the mathematics investigation, Hamilton detected an important defect in one of Laplace’s demonstrations dealing with the composition of forces at the beginning of the work, and he was induced by a friend to write out his remarks, that they might be shown to Dr John Brinkley, then the first Astronomer Royal for Ireland, and an accomplished mathematician. Brinkley seems at once to have perceived the vast talents of young Hamilton, and to have encouraged him in the kindest manner.

Hamilton was 18 when he entered Trinity College, and during his time there achieved great honors. Amongst a number of competitors of more than ordinary merit, he was first in every subject and at every examination. In 1824, he submitted his first paper for publication touching on themes of optics that would later win him and important place in the history of physics. In the same year, he met and hoped to marry Catherine Disney, the daughter of family friends, but lost out to a clergyman who Catherine's mother thought was better situated socially. This created a deep disturbance in Hamilton's emotional life, but he managed to press through in his studies and research. He achieved the rare distinction of obtaining an optime for both Greek and for physics in 1826. He also won awards for his poetry, which would later be commented on, with reserved praise, by the famous bard William Wordsworth, who would become Hamilton's life-long friend.

Early research

Hamilton's first discovery, a method of mathematical investigation equally applicable to optics and dynamics was contained in an early paper which in 1823 Hamilton communicated to Dr Brinkley, by whom, under the title of “Caustics,” it was presented in 1824 to the Royal Irish Academy. It was referred as usual to a committee. Their report, while acknowledging the novelty and value of its contents recommended that, before being published, it should be still further developed and simplified. During the time between 1825 to 1828 the paper grew to an immense bulk, principally by the additional details which had been inserted at the desire of the committee. But it also assumed a much more intelligible form, and the features of the new method were now easily to be seen. Hamilton himself seems not until this period to have fully understood either the nature or importance of optics, as later Hamilton had intentions of applying his method to dynamics.

Hamilton as astronomer

Hamilton's career as a student was cut short by his appointment to the Andrews Professorship of Astronomy in the University of Dublin, vacated by 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, authorized one of their number, who was Hamilton's personal friend, to urge Hamilton to become a candidate, a step which Hamilton's modesty had prevented him from taking. Thus, when barely twenty-two, Hamilton was established at the Dunsink Observatory, near Dublin.

Hamilton was not specially fitted for the post, for although he had a profound acquaintance with theoretical astronomy, he had paid but little attention to the regular work of the practical astronomer. And it must be said that Hamilton’s time was better employed in original investigations than it would have been had he spent it in observations made even with the best of instruments.

Least action and conical refraction

In 1827, Hamilton presented a theory that provided a single function that brings together mechanics, optics and mathematics. The Royal Irish Academy paper was finally entitled “Theory of Systems of Rays,” and the first part was printed in 1828 in the Transactions of the Royal Irish Academy. It is understood that 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.

The third portion of his "Theory of Systems of Rays" appeared in 1832. It was in this paper that he predicted "conical refraction," based on his formulation of optics using the principle of varying action. There was some challenge mounted to his being the first to uncover this phenomeon, but it was later admitted that Hamilton had taken the final and necessary step toward this discovery.

It its contribution to theoretical dynamics, Hamilton's work was the most important since those of Sir Isaac Newton and Joseph Louis Lagrange. C. G. J. Jacobi and other mathematicians have extended Hamilton's processes, and have thus made extensive additions to our knowledge of differential equations.

In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, he 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 corresponding member of the Academy of St Petersburg. These are the few salient points (other, of course, than the epochs of Hamilton's more important discoveries and inventions presently to be considered) in the uneventful life of Hamilton.

Quaternions

The other great contribution made by Hamilton to mathematical science was the introduction of quaternions, beginning in 1843.

Hamilton was looking for ways of extending complex numbers, which are expressed as the sum of a real and imaginary (multiple of the square root of one) component. Hamilton could not extend this branch of mathematics to three dimensions: in fact later mathematicians showed that this would be impossible. Eventually Hamilton tried four dimensions and, by doing so, created quaternions. According to the story Hamilton told, on October 16 Hamilton was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

$i^{2}=j^{2}=k^{2}=ijk=-1$

suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge.) Since 1989, the National University of Ireland, Maynooth has organized a pilgrimage, where mathematicians take a walk from Dunsink observatory to the bridge where, unfortunately, no trace of the carving remains, though a stone plaque does commemorate the discovery. He then developed the rules for multiplication and division of quaternions, which had aluded him for many years.

The quaternion involved abandoning commutativity, a radical step for the time. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. 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.

In 1852, Hamilton introduced quaternions as a method of analysis. His first great work is Lectures on Quaternions (Dublin, 1852). Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research.

He popularized quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was just short of being completed at the time of his death.

Peter Guthrie Tait among others, advocated the use of Hamilton's quaternions. However, controversy about their use grew in the late 1800s. Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (from developers like Oliver Heaviside and J. Willard Gibbs, because quaternions provide superior notation. While this is undeniable for four dimensions, quaternions cannot be used with arbitrary dimensionality (though extensions like Clifford algebras can). Vector notation largely replaced the "space-time" quaternions in science and engineering by the mid-20th century.

Today, the quaternions are in use by 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 many quaternion transformations is more numerically stable than combining many matrix transformations. 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.

Death and afterwards

Hamilton retained his faculties unimpaired to the very last, and steadily continued till within a day or two of his death, which occurred on 2 September 1865, the task of finishing the “Elements of Quaternions” which had occupied the last six years of his life.

Honors

Hamilton is recognized 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 NUI Maynooth and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann, Andrew Wiles and Timothy Gowers have all spoken. 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 TCD.

Character and legacy

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.

Other accomplishments

William Rowan Hamilton's mathematical career included the study of geometrical optics, 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.

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 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.

The Hamiltonian

Hamilton also contributed an alternative formulation of the mathematical theory of classical mechanics. While adding no new physics, this formulation, which builds on that of Joseph Louis Lagrange, provides a more powerful technique for working with the equations of motion. Both the Lagrangian and Hamiltonian approaches were developed to describe the motion of discrete systems, were then extended to continuous systems and in this form can be used to define vector fields. In this way, the techniques find use in electromagnetic, quantum relativity theory and field theory.

Commemorations of Hamilton

• Hamilton's equations are a formulation of classical mechanics.
• Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and a term from graph theory. It can be seen as the Quantum Hamiltonian.

Publications

• Hamilton, William Rowan (Royal Astronomer Of Ireland), "Introductory Lecture on Astronomy." Dublin University Review and Quarterly Magazine Vol. I, Trinity College, January 1833.
• Hamilton, William Rowan, "Lectures on Quaternions." Royal Irish Academy, 1853.
• David R. Wilkins's collection of Hamilton's Mathematical Papers.

References

ISBN links support NWE through referral fees

• Sir William Rowan Hamilton by Thomas Hankins, 1980 published by The Johns Hopkins University Press, 474 pages. Primarily biographical but covers the math and physics Hamilton worked on in sufficient detail to give a flavor of the work.
• Wilkins, David R., Sir William Rowan Hamilton. School of Mathematics, Trinity College, Dublin.
• Wolfram Research's William Rowan Hamilton
• Cheryl Haefner's Sir William Rowan Hamilton

External links

• MacTutor's Sir William Rowan Hamilton. School of Mathematics, University of St Andrews.
• 1911 Britannica Hamilton
• Hamilton Trust
• The Hamilton year 2005 web site
• The Hamilton Mathematics Institute, TCD
• Hamilton Institute

See also

• List of people on stamps of Ireland
• Tarik O'Regan (great, great, great grandson)
• Against the Day

Credits

New World Encyclopedia writers and editors rewrote and completed the Wikipedia article in accordance with New World Encyclopedia standards. This article abides by terms of the Creative Commons CC-by-sa 3.0 License (CC-by-sa), which may be used and disseminated with proper attribution. Credit is due under the terms of this license that can reference both the New World Encyclopedia contributors and the selfless volunteer contributors of the Wikimedia Foundation. To cite this article click here for a list of acceptable citing formats.The history of earlier contributions by wikipedians is accessible to researchers here:

• William Rowan Hamilton  ^history

The history of this article since it was imported to New World Encyclopedia:

• History of "William Rowan Hamilton"

Note: Some restrictions may apply to use of individual images which are separately licensed.