Sir William Rowan Hamilton (August 4th 1805 - September 2nd 1865) is a mathematician, physicist and astronomer Irish. It is known for its discovery of the Quaternion S, but it also contributed to the development of the Optique, the Dynamique and the Algèbre. Its research appeared important for the development of the quantum Mécanique.

Mathematical work of Hamilton includes the study of the geometrical Optique, the adaptation of the dynamic methods to the optical systems, the application of the quaternions and of the Vecteur S with the problems of geometrical Mécanique and S, possibilities of polynomial resolution of the equations S, in particular the general equation of the fifth degree, the linear operators, of which it proves a result concerning these operators in the space of the quaternions and who is a special case of the Théorème of Cayley-Hamilton.

Biography

Youth

Hamilton was born with Dublin, his/her father is Archibald Hamilton. One of the branches of the Scottish family to which it belongs was installed in the north of Ireland at the time of Jacques I {{er}} of Scotland, which sometimes gave the impression that Hamilton was Scottish. Hamilton is educated by James Hamilton, a priest Anglican who was his uncle.

Child prodigy; its genius first of all proved in its capacity to learn the Langue S. At age the 7 years, it already made considerable progresses in Hebrew and, at the 13 years age, under the direction of his uncle who is linguist, it already acquired as many languages as he has years.

These languages were, in addition to the traditional and modern European languages, the Persan, the Arab , the Hindousthânî, the Sanskrit and the Malayan. Although until the end of its life it retained much of its singular training Persan and Arabic, that it reads in the text between two more difficult tasks, it for a long time gave up their study, and the practice simply to release itself.

Hamilton belongs to small but brilliant school of mathematicians associated with the Trinity College with Dublin, where it passed all its life. He studies the traditional ones and sciences and is named professor of Astronomie in 1827, before even being graduate.

Mathematical studies

It seems that Hamilton studies mathematics without any kind of assistance and its research thus does not form part of any school , unless it is not considered that they do not form a school alone.

At the twelve years age Hamilton Zerah Colburn the young person American wonder meets and they mutually test their arithmetic skill at the time of competitions. It seems that pushes Hamilton to increase its knowledge in mathematics.

Two years before, it had fallen accidentally on a copy from the Éléments from Euclide which it devoured greedily. He then undertook the study of the Arithmetica universalis of Newton which is its introduction to the analyzes mathematical modern.

Quickly, it launched out in the reading of the Principia and at sixteen years it controlled of it a great part like that of some more modern works on the analytical Geometry and the differential Calculus.

During this period, Hamilton is engaged for its entry with the Trinity College of Dublin and must thus devote part of its time to the Classiques. At the time of the summer 1822, then seventeen years old, it begins a systematic study of the Celestial mechanics of Laplace. Nothing can better be appropriate to call upon mathematical capacities like those of Hamilton; indeed this philosopher's stone of Laplace is rich in varied but so new analytical processes and requires a neat and often hard study.

It is in this profitable effort to open this box with the treasures that the spirit of Hamilton receives its final hardening.

As from this time, Hamilton seems to be devoted almost entirely to research in mathematics, although it keeps up to date perfectly with the advances in knowledge, in the United Kingdom and abroad. When it detects an important error in one of the Démonstration S of Laplace, a friend pushes it to write his remarks, which could be announced to John Brinkley then the first Astronomer Royal for Ireland and accomplished mathematician. Brinkley seems to distinguish immediately the talents of the young person Hamilton and encourages it in the most pleasant way.

The career of Hamilton at the university is exceptional. Among a certain number of competitors to the merits more than ordinary, it is first in each subject and with each examination. For example, it carries out the rare prowess to obtain the maximum note at the same time in Greek and Physique.

It was awaited that Hamilton gains the two gold medals with the final examination but its career of student was shortened by an event without precedent, its nomination like Astronomer Royal of Ireland, posts become vacant following the nomination of Brinkley to the episcopate, like its nomination little time after at the post of professor of astronomy of Trinity College .

The station was not offered to him exactly, as that was sometimes affirmed, but the voters, had met, tackled the subject, and had authorized one of them, which was the personal friend of Hamilton, to invite it to stand as a candidate; an initiative which the modesty of Hamilton had prevented it from taking.

At twenty-two years and still without diploma, Hamilton is established with the Observatoire Dunsink, close to Dublin. Hamilton, is not appropriate especially for this station, because although having a thorough knowledge of the theoretical Astronomie it carried only little of attention to the normal work of an astronomer.

As its time is used better for original research than to carry out observations, even with the best of the instruments, the authorities of the university which had chosen it want that it as well as possible devotes its time for the advance of science, without being attached a particular branch. If Hamilton had wanted to be devoted to practical astronomy, there is no doubt that the university of Dublin would have provided him instruments and an adequate team of assistants.

In 1835, then secretary of the meeting of the British Association which is held this year in Dublin, it is anobli by the Lord Lieutenant. This same year, he is prize winner of the Royal Medal. But of the honors much more important quickly follow one another, among which its election in 1837 in the place of president of the Royal Irish Academy, and the rare distinction to be member corresponding of the Académie of Saint-Petersbourg.

Work

Optics and dynamics

William Hamilton contributed important shares in Mathematical physics, especially in Optique and Dynamique.

Its first discovered appears in one of the first writing that it communicates to Dr. Brinkley in 1823 and who under the title of Caustics (caustic) is introduced in 1824 to the Royal Irish Academy. The article, as usual, is subjected at a reading panel. Their report/ratio, although recognizing the innovation and the value of its contents, recommends that before publication, the article initially is developed and simplified.

Between 1825 and 1828, the article becomes considerable extensive, mainly by the addition of details required by the committee; but it is also made much more comprehensible and the characteristics of the new method become perfectly apparent.

In 1827, it presents a theory according to which a single function links Mécanique, optics and mathematics and which helped to establish the undulatory Théorie of the light. The article is finally named Theory off Systems off Rays (April 23rd 1827, Théorie of the systems of rays ) and the first part is published in 1828 in the Transactions off the Royal Irish Academy .

The variational Principle, also called principle of Hamilton , is the essential component of these articles. This principle which, reformulated by Jacobi, leads to an alternative formulation of the traditional mechanical ; it is currently known under the name of Hamiltonian Mécanique .

This formulation, as the Lagrangian Mechanical on which it is based, is very mathematical and again does not bring anything to physics but provides a more powerful method to solve the equations of the movement. Mechanics Lagrangian and Hamiltonian was developed to describe the movement of discrete systems; they were extended to the continuous systems using of the Champ S. In this form, they are used in electromagnetism and quantum Mécanique or relativity.

Quaternions

The other great contribution of Hamilton, in pure mathematics this time, is the invention of the Quaternion S. He discovers them in 1843 whereas he seeks a way of extending the complex numbers to dimensions superiors to 2.

He does not find anything in dimension 3, but dimension 4 the conduit with the quaternions. According to the history told by Hamilton itself, the October 16th, whereas he walks with his wife along the Royal Channel to Dublin, the solution came to him suddenly to mind: $i^2\; =\; j^2\; =\; k^2\; =\; ijk\; =\; -1$; it then hastens to engrave this equation on the Brougham Bridge (currently Broom Bridge ).

Since 1989, the National University off Ireland de Maynooth organizes a pilgrimage where mathematicians (in particular Murray Freezing-Mann in 2002 and Andrew Wiles in 2003) traverse the way from the observatory of Dunsink to the bridge, where unfortunately one can see no trace of this inscription.

The introduction of the quaternions at the time has a consequence, considered as essential: the abandonment of the Commutation. With the quaternions, Hamilton invents also the word Vecteur : indeed, it describes the quaternions like an ordered succession of 4 real numbers and calls the first the part Scalaire and the three others the part vector .

Its results on the quaternions are exposed in Lectures one Quaternions (Dublin, 1852) but Hamilton also tried to popularize those in several works, whose last, Elements off Quaternions , comprises 800 pages and was published shortly after its death.

The use of the quaternions is the controversy object.

Hamilton thought that the quaternions would have a great influence like instrument of research and Peter Guthrie TAIT, among others, pleads for their use. Some of the partisans of Hamilton are opposed to the vectorial algebra, developed in particular by Oliver Heaviside and Willard Gibbs, because the quaternions, according to them, offer a better notation. Even if that is debatable for dimension 4, the quaternions cannot be used in an unspecified number of dimensions (although extensions, like the Octonion S or the algebras of Clifford exist).

Also, although the quaternions allow certain elegant and concise demonstrations, the quaternions are seldom used by the mathematicians of the 21e century; the vectorial notation having replaced the quaternions in sciences and engineering lasting half of the 20th century. Let us note all the same that the unit quaternions are the subject of an intensive use in branches like the Synthèse of image and the Animation, the treatment of the signal and orbital mechanics, mainly in the handling of rotations or the orientations.

Others

Hamilton was accustomed to letting mature its ideas before laying down them on paper. The discoveries, articles and ouvages mentioned previously would have already been enough to fill long and hard life. But even without mentioning its enormous quantity of volumes being now in Trinity College of Dublin, this work constitutes only part of those which it published.

Hamilton studied lengthily what relates to the algebraic solutions of the equations of the fifth degree. The results of this research, used inter alia by Niels Abel and George Jerrard, are another of its contributions to science. He studied also in-depth the solutions (in particular by numerical Approximation) of certain classes of differential equations whose only some elements were published, by intervals, in the Philosophical Magazine . One owes him also the very clever invention of the Hodographe.

Hamilton also maintained a very bulky correspondence. Often, only one of its letters occupied fifty or hundred pages with the tightened writing, all devoted to the meticulous considerations of each detail of a particular problem. It was, indeed, one of the characteristics of its spirit to be able to be satisfied general comprehension of a question; it never gave up a problem as long as it had not studied it in its least details. Hamilton was also very courteous to answer requests for assistance concerning the study of its work, and that even when that took most of its time to him. It was also extremely precise and difficult to satisfy with regard to the care taken to the completion of its work for publication and it is probably for this reason which it published only if few in comparison of extended of its research.

Hamilton kept its intact faculties until the end and, at one or two days of its death, it still assiduously continued the development of its Elements off Quaternions task which had occupied the six last years of its life.