Sir William Rowan Hamilton (4
August 1805 – 2 September 1865) was an
Irish physicist,
astronomer, and
mathematician, who made important
contributions to
classical
mechanics,
optics, and
algebra. His studies of mechanical and optical
systems led him to discover new mathematical concepts and
techniques. His greatest contribution is perhaps the 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 mathematics, he is perhaps best known as the
inventor of
quaternions. Hamilton is said
to have shown 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.”
Life
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.
Early life
Hamilton
was the fourth of nine children born to Sarah Hutton (1780–1817)
and Archibald Hamilton (1778–1819), who lived in Dublin at 38
Dominick Street. Hamilton's father, who was from Dunboyne, worked as a
solicitor. By 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 Talbots Castle. His
uncle soon discovered that Hamilton had a remarkable ability to
learn languages.At a young age, Hamilton displayed an uncanny
ability to acquire languages (although this is disputed by some
historians, who claim he had only a very basic understanding of
them). At the age of seven he had already made very 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, as well as
Persian,
Arabic,
Hindustani,
Sanskrit, and even
Marathi and
Malay. He retained much of his knowledge of
languages to the end of his life, often reading Persian and Arabic
in his spare time, although he had long stopped studying languages,
and used them just for relaxation.
Hamilton
later attended Westminster
School with Zerah
Colburn. He was part of a small but well-regarded
school of mathematicians associated with Trinity College,
Dublin, where he spent his life. He studied both
classics and science, and was appointed Professor of
Astronomy in 1827, prior to his graduation.
Optics and mechanics
Hamilton made important contributions to
optics and to
classical mechanics. His first discovery
was in an early paper that he communicated in 1823 to Dr. Brinkley,
who presented it under the title of "
Caustics" in 1824 to
the
Royal Irish Academy. It was
referred as usual to a committee. While their report acknowledged
its novelty and value, they recommended further development and
simplification before publication. Between 1825 to 1828 the paper
grew to an immense size, mostly by the additional details which the
committee had suggested. But it also became more intelligible, and
the features of the new method were now easily to be seen. Until
this period Hamilton himself seems to not have fully understood
either the nature or importance of optics, as later he intended to
apply his method to dynamics.
In 1827, Hamilton presented a theory of a single function, now
known as
Hamilton's
principal function, that brings together mechanics, optics, and
mathematics, and which helped to establish the wave theory of
light. He proposed for 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 is still known by its original name, "
conical refraction".
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. The common thread
running through all this work is Hamilton's principle of
“
Varying Action“. Although it is based on the
calculus of variations and may be
said to belong to 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 revealed much deeper mathematical structure
than had been previously understood, in particular the symmetry
between momentum and position. Paradoxically, the credit for
discovering the quantity now called the
Lagrangian and
Lagrange's equations belongs to
Hamilton. Hamilton's advances enlarged greatly the class of
mechanical problems that could be solved, and they represent
perhaps the greatest addition which
dynamics had received since the work of
Isaac Newton and
Lagrange.
C.
G. J.
Jacobi,
Joseph Liouville,
Jean Gaston Darboux,
Henri Poincare,
Kolmogorov,
V.
I. Arnold,
and other scientists have extended Hamilton's work, and have thus
made extensive additions to our knowledge of
mechanics and
differential equations.
While Hamilton's reformulation of classical mechanics is based on
the same physical principles as the mechanics of Newton and
Lagrange, it provides a powerful new technique for working with the
equations of motion. More importantly, both the
Lagrangian and
Hamiltonian approaches which were
initially developed to describe the motion of
discrete systems, have proven critical to
the study of continuous classical systems in physics, and even
quantum mechanical systems. In this way, the techniques find use in
electromagnetism,
quantum mechanics,
quantum relativity theory, and
field theory.
Mathematical studies
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". Not only was
Hamilton an expert as an
arithmetic
calculator, but he seems to have occasionally had fun in working
out the result of some calculation to an enormous number of decimal
places.
At
the age of twelve Hamilton engaged Zerah Colburn, the American "calculating
boy", who was then being exhibited as a curiosity in
Dublin, and did not always lose. Two years before, he had
stumbled into a
Latin copy of
Euclid, which he eagerly devoured; and at twelve
Hamilton studied
Newton’s
Arithmetica Universalis. This
was his introduction to modern
analysis. Hamilton soon began to read
the
Principia,
and at sixteen Hamilton had mastered a great part of it, as well as
some more modern works on
analytical
geometry and the
differential
calculus.
Around
this time Hamilton was also preparing to enter Trinity
College, Dublin, and therefore had to devote some time to
classics. In mid-1822 he began a systematic study of
Laplace's
Mécanique Céleste.
From that time 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
Astronomer Royal for
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 career at College was perhaps unexampled. Amongst a
number of extraordinary competitors, he was first in every subject
and at every examination. He achieved the rare distinction of
obtaining an
optime for both
Greek and for
physics.
Hamilton might have attained many more such honours (he was
expected to win both the
gold medals at
the degree examination), if his career as a student had not been
cut short by an unprecedented event.
This was Hamilton’s
appointment to the Andrews Professorship of
Astronomy in the University of Dublin, vacated by Dr. 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
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.
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.
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 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.
Quaternions
The other great contribution Hamilton made to mathematical science
was his discovery of
quaternions in
1843.
Hamilton was looking for ways of extending
complex numbers (which can be viewed as
point on a 2-dimensional
plane) to higher spatial dimensions. He
could not do so for 3 dimensions, and it was later shown that it is
impossible. Eventually Hamilton tried 4 dimensions and 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
- \displaystyle i^2 = j^2 = k^2 = ijk = -1
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), for fear
he would forget it. Since 1989, the National
University of Ireland, Maynooth has organized a pilgrimage, where 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.
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 published
shortly after his death.
Peter Guthrie Tait among others,
advocated the use of Hamilton's quaternions. They were made a
mandatory examination topic in Dublin, and for a while they were
the only advanced mathematics taught in some Americanuniversities.
However, controversy about the use of quaternions 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
Josiah 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 had
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.
Other originality
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.
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.
Death and afterwards
Hamilton retained his faculties unimpaired to the very 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 September 2, 1865, following a severe attack of
gout.
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, 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 TCD.
A commemorative coin was issued by the Central Bank of Ireland in
his honour.
Commemorations of Hamilton
Quotations
- "Time is said to have only one dimension, and space to have
three dimensions. ... The mathematical quaternion partakes of both
these elements; in technical language it may be said to be 'time
plus space', or 'space plus time': and in this sense it has, or at
least involves a reference to, four dimensions. And how the One of
Time, of Space the Three, Might in the Chain of Symbols girdled
be."—William Rowan Hamilton (Quoted in Robert Percival Graves' "Life of
Sir William Rowan Hamilton" (3 vols., 1882, 1885, 1889))
- "He used to carry on, long trains of algebraic and arithmetical
calculations in his mind, during which he was unconscious of the
earthly necessity of eating; we used to bring in a ‘snack’ and
leave it in his study, but a brief nod of recognition of the
intrusion of the chop or cutlet was often the only result, and his
thoughts went on soaring upwards."—William Edwin Hamilton (his elder
son)
See also
Notes
References
- , 474 pages—Primarily biographical but covers the math and
physics Hamilton worked on in sufficient detail to give a flavor of
the work.
External links
Publications