William rowan hamilton

William Rowan Hamilton (United Kingdom: /ˈwɪlɪəm ˈɹəʊən ˈhæmɪltn̩/; Dublin, 4 August 1805-ibid., 2 September 1865) was an Irish mathematician, physicist, and astronomer, who made important contributions to the development of optics, dynamics, and algebra. His discovery of the quaternion, along with his systematization of dynamics, are his best known works. This last work would be decisive in the development of quantum mechanics, where a fundamental concept called the Hamiltonian is named after him.

Semblance

Early Years

Hamilton was the fourth of nine children born to Sarah Hutton (1780-1817) and Archibald Hamilton (1778-1819), who lived at 29 Dominick Street in Dublin. Hamilton's father, who was Dubliner, worked as a lawyer. At 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 the town of Talbots Castle, in Trim, County Meath.

Hamilton is said to have displayed immense talent at a very young age. Hamilton's predecessor as Astronomer Royal of Ireland and later Bishop of Cloyne, Dr John Brinkley, commented on Hamilton when he was 18: This young man, I do not say "will be", I say & #34;es", the first mathematician of his time.

His uncle observed that Hamilton, from an early age, had displayed an uncanny ability to learn languages (although this claim is disputed by some historians, who claim that he had only a very basic understanding of them). By the age of seven, he had already made considerable progress with Hebrew, and by the time he was thirteen, under the supervision of his uncle (a linguist), he had acquired knowledge of almost as many languages as he was years old (classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, and even Marathi and Malay). He retained much of his knowledge of languages until the end of his life, often reading Persian and Arabic in his spare time, though he had long since given up studying languages and used them only to relax.

In September 1813, the American prodigy calculator Zerah Colburn was exhibiting in Dublin. Colburn was 9 years old, a year older than Hamilton. The two faced each other in a mental arithmetic contest, with Colburn being the clear winner.As a result of Colburn's defeat, Hamilton spent less time studying languages and more time studying mathematics.

Education

Hamilton was part of a small but well-regarded school of mathematics associated with Trinity College Dublin, which he entered at the age of 18. The university awarded him two Optimes, or Out of the Normal marks. He completed both studies classics such as mathematics (bachelor of arts in 1827, and master's degree in 1837). While still a student, he was appointed Andrews Professor of Astronomy and Astronomer Royal of Ireland, he subsequently settled at Dunsink Observatory, to which he remained attached for the rest of his life.

Personal Life

While attending Trinity College, Hamilton proposed to a friend's sister, who turned him down. Hamilton, being a sensitive young man, became ill and depressed, even on the brink of suicide. He was turned down again in 1831 by Ellen de Vere, sister of the poet Aubrey Thomas de Vere (1814-1902). Eventually Helen Marie Bayly, daughter of a rural preacher, accepted his proposal and they married in 1833. The couple had three children: William Edwin Hamilton (b. 1834), Archibald Henry (b. 1835)), and Helen Elizabeth (b. 1840). Bayly proved to be extremely pious, shy, and reserved; What's more, she suffered from a chronic illness, so Hamilton's married life presumably must not have been easy.

Perhaps the most memorable moment of his life was when, according to his own account, the structure of the quaternionic numbers came to his head like a flash of lightning. Evidently Hamilton had been thinking about this problem for a long time; but, be that as it may, one day in 1843 he was walking with his wife on Brongham Bridge, which crosses the Royal Dublin Canal, when he suddenly understood the structure of quaternions. Immediately afterwards he engraved with the point of his knife, on a stone of the bridge, the happy idea (the inscription has not been preserved today).

Death and Legacy

The mathematician kept his mental faculties intact until the end of his life, and he constantly continued the task of finishing the "Elements of quaternions" that had occupied the last six years of his life. He died on September 2, 1865, after a severe attack of gout.He is buried in Mount Jerome Cemetery in Dublin.

Hamilton is recognized as one of Ireland's foremost scientists, and as the nation becomes more aware of its scientific heritage, he is increasingly celebrated. It is said that he was allowed to step on the University lawn, something totally prohibited. This fact walks between reality and fiction. Possibly it happened that, absorbed in his musings, he neglected this prohibition and accidentally walked through the gardens, although no one in all Ireland would have dared to interrupt or admonish him. This anecdote surely serves to give an idea of Hamilton's category as one of the great mathematicians of his time and of history.

The Hamilton Institute is dedicated to research in applied mathematics at Maynooth University, and the Royal Irish Academy holds an annual public lecture commemorating Hamilton featuring, among others, Murray Gell-Mann, Frank Wilczek, Andrew Wiles and William Timothy Gowers. The year 2005 was the 200th anniversary of Hamilton's birth and the Irish government designated it the 'Year of Hamilton, celebrating Irish science'. Trinity College Dublin marked the year with the opening of the William Rowan Hamilton Institute.

Ireland issued two commemorative stamps in 1943 to celebrate the centenary of the quaternions' announcement. The Central Bank of Ireland struck a commemorative silver €10 coin in 2005 to mark 200 years since their birth.

The newest maintenance workshops on the Dublin Tram System (LUAS) bear his name.

Astronomy

In his youth, Hamilton owned a telescope, and became an expert in calculating celestial phenomena, such as determining the visibility of lunar eclipses. Having received extremely high marks in both Classics as in Science, it was not too unusual that, on June 16, 1827, just 21 years old and still a student, he was elected Astronomer Royal of Ireland and took up residence 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 consisted mainly of measuring the positions of the stars, which was not very interesting to a mathematical mind. But the main reason he eventually turned over regular observing entirely to his astronomy assistant, Charles Thompson, was that Hamilton frequently suffered from illness after taking up observing.

Today, Hamilton is not recognized as a great astronomer, although during his lifetime he was recognized as such. His introductory lectures on astronomy were famous; besides his students, they attracted many scholars and poets, and even ladies; in those days a remarkable feat.Poet Felicia Hemans penned her poem "The Lonely Student's Prayer"; after listening to one of her lectures.

Physics

Hamilton made important contributions to optics and classical mechanics. The first discovery of his was published in an early paper which he communicated in 1823 to Dr. Brinkley, who presented it in 1824 under the title of & # 34;Caustics & # 34; to the Royal Irish Academy. He referred as usual to a committee. While the issued report recognized its novelty and value, it was recommended that it should be further developed and simplified prior to publication. Between 1825 and 1828, the document had grown to an immense size, mainly because of the additional details that the committee had suggested. But it also became more intelligible, and the features of the new method were now easily seen. Until this period, Hamilton himself seems not to have fully understood either the nature or the importance of optics, since he later claimed to apply his method to dynamics.

In 1827, he presented a theory of a single function, now known as the Hamilton-Jacobi equation, which unites mechanics, optics, and mathematics, and which helped establish the wave theory of light. 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 eventually titled &# 34;Theory of Ray Systems" (23 April 1827), and the first part was printed in 1828 in the "Transactions of the Royal Irish Academy". The most important contents of the second and third parts appeared in the three voluminous supplements (to the first part) that were published in the same Transactions, and in the two articles "On a general method in dynamics", which appeared in Philosophical Transactions in 1834 and 1835. In these articles, Hamilton developed his great principle of "variable action". The most notable 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 is still known by its original name, "conical refraction".

The transition from optics to dynamics in the application of the method of "variable action" was made in 1827, and reported to the Royal Society, in whose Transactions of 1834 and 1835 there are two papers on the subject, which, as with "ray systems", assume a mastery over symbols and an almost unmatched fluency in mathematical language. The common thread of all this work is the principle of "variable action" of Hamilton. Although it is based on the calculus of variations and can be said to belong to the general class of problems included in the principle of least action that had been previously studied by Pierre Louis Maupertuis, Leonhard Euler, Joseph-Louis Lagrange, and others, the analysis of Hamilton revealed a much deeper mathematical structure than had been previously understood, particularly regarding the symmetry between momentum and position. Paradoxically, the credit for discovering the operator now called Lagrangian and Lagrangian mechanics belongs to Hamilton. His advances greatly expanded the class of mechanical problems that could be solved, and represent perhaps the greatest contribution to dynamics since the work of Isaac Newton and Joseph-Louis Lagrange. Many scientists, including Liouville, Jacobi, Darboux, Poincaré, Andrei Kolmogorov, and Arnold, have expanded on Hamilton's work, thus extending our knowledge of mechanics and differential equations and forming the basis of symplectic topology.

Although Hamiltonian mechanics is based on the same physical principles as Newtonian and Lagrange mechanics, it provides a powerful new technique for working with the equations of motion. More importantly, the Lagrangian and Hamiltonian approaches, which were initially developed to describe the motion of discrete systems, have proven to be fundamental for the study of continuous classical systems in physics, and even in quantum mechanical systems. In fact, the techniques find use in electromagnetism, quantum mechanics, quantum relativity theory, and quantum field theory. In the Irish Dictionary of Biographies, David Spearman wrote:

Despite the importance of its contributions to algebra and optics, posterity gives it the greatest reputation for its dynamics. The formulation that he designed for classical mechanics turned out to be equally adequate for quantum theory, whose development facilitated. Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find in this the most natural medium for its description and development, and the function now universally known as hamiltonian, is the starting point for calculus in almost any area of physics.

Math

Hamilton's mathematical studies seem to have been carried out and reached their full development without any help, and the result is that his writings do not belong to any "school&# 3. 4; in particular. Hamilton was not only an expert in arithmetic calculation, but sometimes it seems that he amused himself by calculating the result of some calculations with a huge number of decimal places. At the age of eight, Hamilton competed with Zerah Colburn, the "calculating kid" American, then on display as a curio in Dublin. Two years later, at the age of ten, Hamilton stumbled across a Latin copy of Euclid's Elements, which he enthusiastically devoured; and at twelve he studied Newton's Arithmetica Universalis. This was his introduction to modern analysis. Hamilton soon began to read the Principia, and by the age of sixteen had mastered much of it, as well as some more modern work on analytic geometry and differential calculus.

At this time, Hamilton was also preparing to enter Trinity College, Dublin, so he had to devote some time to the classics. In the middle of 1822 he began a systematic study of the & # 34; Mécanique Céleste & # 34; by Pierre-Simon Laplace.

From that time on, Hamilton seems to have devoted himself almost entirely to mathematics, although he always remained familiar with the state of the science both in Britain and abroad. Hamilton found a major flaw in one of Laplace's proofs, and a friend induced him to write his comments so they could show them to Dr. John Brinkley, then Ireland's first Astronomer Royal, and an accomplished mathematician. Brinkley seems to have sensed Hamilton's talent at once, and encouraged it in the kindest way.

Hamilton's college career was perhaps unprecedented. Among a series of extraordinary competitors, he was first in all subjects and in all exams. He achieved the rare distinction of earning a wrangler in both Greek and Physics. Hamilton could have garnered many more such honors (he was expected to win both gold medals in the graduation exam) if his career as a student had not been cut short by an unprecedented event. This was Hamilton's appointment as Andrews Professor of Astronomy at Dublin University, vacated by Dr Brinkley in 1827. The chair was not offered to him directly, as has sometimes been claimed, but the electors, having met and discussed on the subject, they authorized Hamilton's personal friend (also an elector) to urge him to become a candidate, a step Hamilton's modesty had prevented him from taking. Thus, when he was barely 22 years old, Hamilton settled at the Dunsink Observatory, near Dublin.

Hamilton was not particularly suitable for the position, because although he was well acquainted with theoretical astronomy, he had paid little attention to the regular work of the practical astronomer. Hamilton's time was better spent on original research than on observations made with even the best instruments. The university authorities who elected him to the chair of astronomy intended that Hamilton spend his time to the best of his ability for the advancement of science, without being tied to any particular branch. If Hamilton had devoted himself to practical astronomy, the University of Dublin would surely have provided him with instruments and an adequate staff of assistants.

He was twice awarded the Cunningham Medal of the Royal Irish Academy. 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 He would win it again in 1848.

In 1835, while secretary of the meeting of the British Association for the Advancement of Science held that year in Dublin, he served as a gentleman sent by the Lord Lieutenant of Ireland. He quickly received other honors, including his election in 1837 to the presidency of the Royal Irish Academy, and the rare distinction of being made a corresponding member of the Russian 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 at the top of its list.

Quaternions

Cuaternion commemorative plaque at Broom Bridge

Hamilton's other major contribution to mathematical science was his discovery of quaternions in 1843. By 1840, however, Olinde Rodrigues had already achieved a result that equaled his discovery in all but name.

Hamilton was looking for ways to extend complex numbers (which can be seen as points on a two-dimensional plane) to higher spatial dimensions. He couldn't find a useful three-dimensional system (in modern terminology, he couldn't find a real three-dimensional division ring), but by working with four dimensions he created quaternions. According to Hamilton himself, on October 16 he was walking along the Royal Canal in Dublin with his wife, when suddenly the solution in the form of an equation occurred to him:

i² = j² = k² = ijk = −1

and quickly carved this equation using his penknife on the side of the nearby Broom Bridge (which Hamilton named Brougham Bridge). This event marked the discovery of the quaternion group.

The Taoiseach (Irish chief executive) Éamon de Valera, a mathematician and student of quaternions, would inaugurate years later a commemorative plaque attached to the bridge on November 13, 1958. Since 1989, the National University of Ireland, Maynooth, has organized a walk called the Hamilton Walk, in which mathematicians take a walk from the Dunsink Observatory to the bridge, where there is no trace of the marks originally made by Hamilton, although a stone plaque commemorates the discovery.

The quaternion implied abandoning commutativity, a radical step for the time. Not only this, Hamilton also invented the dot and cross products of vector algebra, the product of quaternions being the cross product minus the dot product. Hamilton also described the quaternion as an ordered set of four elements of real numbers, and described the first element as the "scalar" part of the quaternion. and the remaining three as the "vector" part. He 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 through the introduction of complex coefficients. By the time his work was assembled in 1853, the book "Lectures on Quaternions" had "formed the subject of successive courses of lectures, delivered in 1848 and later years, in the Halls of Trinity College, Dublin & # 34;. Hamilton confidently stated that quaternions would have a powerful influence as a research tool.

When he died, he was working on a definitive statement of quaternion science. His son, William Edwin Hamilton, submitted the "Elements of Quaternions," a considerable volume of 762 pages, for publication in 1866. As copies went out of print, a second edition was prepared by Charles Jasper Joly, when the book it was divided into two volumes, the first appearing in 1899 and the second in 1901. The subject index and footnotes in this second edition improved the ease of understanding of the "Elements".

One of the features of the Hamilton quaternion system was the differential operator nabla, which could be used to express the gradient of a vector field or to express the curl. These operations were applied by Maxwell to Michael Faraday's electrical and magnetic studies in Maxwell's Treatise on Electricity and Magnetism (1873). Although the nabla operator is still used, real quaternions fall short as a representation of spacetime. On the other hand, the algebra of biquaternions, in the hands of Arthur W. Conway and Ludwik Silberstein, provided the space-time representation tools of Minkowski and the Lorentz group at the beginning of the 20th century.

Today, quaternions are used in computer graphics, control theory, signal processing, and orbital mechanics, primarily to represent rotations/orientations. For example, it is common for spacecraft control systems to be specified in terms of quaternions, which are also used to telemeter their current state. The reason is that the combination of quaternion transformations is numerically more stable than the combination of many matrix transformations. In control and modeling applications, quaternions do not have a computational singularity (indefinite division by zero) that can occur for the quarter-turn (90 degree) rotations that are achievable by many air, sea, and space vehicles. In pure mathematics, quaternions feature significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry.

Some modern mathematicians believe that Hamilton's work on quaternions was satirized by Lewis Carroll in Alice's Adventures in Wonderland. In particular, the Mad Hatter's tea party was meant to represent the craze of quaternions and the need to return to Euclidean geometry.

Other original works

Hamilton originally matured his ideas before putting pen to paper. The aforementioned discoveries, articles, and treatises might well have formed the entire work of a long and industrious life. But not to mention his enormous book collection, brimming with new and original material, which has been delivered to Trinity College (Dublin), the works mentioned above hardly make up the bulk of what Hamilton has published. He developed the variational principle, which was later reformulated by Carl Gustav Jakob Jacobi. He also introduced the game Icosian or 'Hamilton's puzzle'. which can be solved using the concept of a Hamiltonian path.

Hamilton's extraordinary investigations related to the solution of algebraic equations of the fifth degree, and his examination of the results obtained by N. H. Abel, G. B. Jerrard, and others in their investigations on this subject, constitute another contribution to science. Below is Hamilton's paper on fluctuating functions, a subject which, since the time of Joseph Fourier, has been of immense and ever increasing value in applied mathematics in physics. There is also the extremely ingenious invention of the hodograph. Of his extensive investigations into the solutions (especially by numerical analysis) of certain classes of physical differential equations, only a few items have been published, at intervals, in the Philosophical Magazine.

In addition to all this, Hamilton carried on a voluminous correspondence. Often a single letter from Hamilton would occupy fifty to a hundred or more written pages, all devoted to minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of the Hamiltonian mind that it was never satisfied with a general understanding of a matter; and he went on with the problem until he knew it in all its details. He was always courteous and kind in responding to requests for help in studying his works, even when it must have taken him a long time to comply. He was excessively precise and difficult to please with reference to the final polishing of his own works for his publication; and it was probably for this reason that he published so little in comparison with the scope of his research.

Eponymy

• Hamiltonian mechanics, a formulation of classical mechanics.
• Many other concepts and objects in mechanics, such as Hamilton's principle, Hamilton-Jacobi's equation, or Cayley-Hamilton's theorem, bear Hamilton's name.
• Hamiltonian is the name of a function (classic) and an operator (quantum) in physics, and in a different sense, a term of the theory of graphs.
• The Haitian road is another concept that bears its name.
• The 'Society of Hamilton', an organization of students of the Royal College of Surgeons of Ireland, was founded in its honor in 2004.
• The algebra of quaternions is usually denoted by Hor by H{displaystyle mathbb {H} }In honor of Hamilton.
• The Hamilton building at Trinity College Dublin is named after it.
• The lunar crater Hamilton carries this name in his memory.

Posts

• 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.
• Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
• Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; Longmans, Green & Co..
• David R. Wilkins's collection of Hamilton's Mathematical Papers Archived on 30 May 2013 at Wayback Machine..