William Rowan Hamilton

How to use this course

Every physics student writes Hamilton’s name many times a week. The Hamiltonian operator of quantum mechanics, Hamilton’s equations of motion and Hamilton’s principle are all named for him. Mathematicians know him for the quaternions and for Hamiltonian cycles in graphs. Few readers know that these ideas came from one man working mostly alone at an observatory outside Dublin, and that the mechanics grew directly out of his optics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow his work in order: optics (1824–1833), dynamics (1834–1835), algebra (1833–1843) and quaternions with their offshoots (1843–1865).

Timeline

Dublin and Trim, 1805–1823

William Rowan Hamilton was born in Dublin at midnight between 3 and 4 August 1805. For years he kept 3 August as his birthday and changed to 4 August only in his thirties. His father Archibald was a solicitor whose business often took him away to England. As a small child he was sent to live with his father’s brother, the Rev. James Hamilton, who kept a school at Trim in County Meath. He stayed there through his boyhood.

James Hamilton was a linguist, and he trained his nephew accordingly. The boy read Latin, Greek and Hebrew as a young child and went on to French, Italian, and a list of eastern languages including Persian, Arabic and Sanskrit. The list was the kind of thing his family liked to recite, and later biographers have treated parts of it with caution. Mathematics came later. As a boy he met the American calculating prodigy Zerah Colburn, who was touring Britain, and competed with him at mental arithmetic. Hamilton lost, and the defeat seems to have turned his interest toward mathematics.

Self-taught mathematics

At thirteen he worked through Alexis Clairaut’s Algebra in French. By fifteen he was reading Isaac Newton and Pierre-Simon Laplace. In 1822, aged seventeen, he found an error in Laplace’s Mécanique céleste. The discovery brought him to the attention of John Brinkley, the Royal Astronomer of Ireland. Hamilton was also writing on the geometry of curves and on caustics, the bright curves formed when light reflects from a curved mirror.

Trinity College, 1823–1827

In July 1823 Hamilton entered Trinity College Dublin by examination. His undergraduate record became a Dublin legend. He won the rare grade of optime in classics in his first year, a mark given only a few times in a generation, and in 1826 he received optimes in both classics and science. He also wrote a great deal of poetry, much of it about Catherine Disney, the sister of a college friend. Her family did not favour the match, and in 1825 she married the Rev. William Barlow. The loss marked him for life, and she reappears at the end of his story.

The theory of rays

In 1824 he sent a paper on caustics to the Royal Irish Academy. The referees asked for more development, and he rebuilt it into a general theory. On 23 April 1827 he presented Theory of Systems of Rays to the Academy. Its central idea was that all the properties of an optical system can be read off from a single function, which he called the characteristic function. The first part was printed in the Academy’s Transactions in 1828. Lesson 04 explains it.

A professor before graduation

In 1827 Brinkley left to become Bishop of Cloyne, and his chair, the Andrews Professorship of Astronomy, fell vacant. The college passed over established astronomers and chose Hamilton, then 21 and still an undergraduate. The post carried the title Royal Astronomer of Ireland and a house at Dunsink Observatory, a few miles north-west of Dublin. He took his BA the same year.

Dunsink, 1827–1835

Hamilton had little observing experience, and the appointment was controversial for that reason. He lectured on astronomy with success, but he soon lost interest in observing, and from the start his real work was mathematical. Dunsink was his home for the rest of his life.

Before taking up the post he toured England and the Lake District in 1827 and met William Wordsworth. The two became friends. Hamilton sent the poet his verses, and Wordsworth gently advised him that his gift lay in science. He also came to know Samuel Taylor Coleridge, whose philosophy, with that of Immanuel Kant, shaped his later view of algebra.

The great years

The next eight years were the most productive of his life. Three supplements to the theory of rays extended it to crystals, and on 22 October 1832 the third supplement predicted conical refraction, a new optical effect confirmed by his colleague Humphrey Lloyd within two months (Lesson 05). He turned the same method on mechanics in two essays printed in the Philosophical Transactions in 1834 and 1835 (Lessons 06 and 07). In 1833 he read a paper defining complex numbers as ordered pairs of reals (Lesson 08).

In 1833 he married Helen Bayly, daughter of a clergyman from Nenagh in County Tipperary. They had two sons, William Edwin (1834) and Archibald Henry (1835), and a daughter, Helen Eliza Amelia (1840). In 1835 the British Association met in Dublin, with Hamilton as one of its secretaries. During the meeting the Lord Lieutenant of Ireland knighted him. The same year the Royal Society awarded him a Royal Medal. He was thirty.

Systems of rays and the characteristic function

For the general reader. Light takes the quickest path. Hamilton asked what follows if you know the travel time between every pair of points. He found that this one function, the characteristic function, contains everything about an optical instrument: where rays go, where they focus, and what images look like. It was the first time a whole branch of physics had been reduced to a single generating function.

The characteristic function

Let \(n(\mathbf x)\) be the refractive index of an isotropic medium. Fermat’s principle says that a ray from \(\mathbf x_0\) to \(\mathbf x\) makes the optical path length \(\int n\,ds\) stationary. Hamilton defined

Derivation

Move the endpoint from \(\mathbf x\) to \(\mathbf x+\delta\mathbf x\). The new ray differs from the old one by a small variation. Because the old ray makes \(\int n\,ds\) stationary with fixed ends, the first-order change in the integral comes only from the boundary term. Writing the path as \(\mathbf r(s)\), the variation of \(\int n|\mathbf r'|\,ds\) gives the boundary term \(n\,\mathbf t\cdot\delta\mathbf x\) at the moving end. Hence \(\delta V=n\,\mathbf t\cdot\delta\mathbf x\), which is the first formula. The second follows the same way, with a minus sign because the lower limit moves.

Fix \(\mathbf x_0\). Then \(|\nabla V|=n\), which is the eikonal equation

The rays are the curves orthogonal to the level surfaces \(V=\text{const}\), which are the wavefronts. This is the Malus–Dupin theorem, which Étienne-Louis Malus had proved for a single reflection or refraction and Charles Dupin had extended: a family of rays normal to some surface stays normal to a family of surfaces after any number of reflections and refractions. In Hamilton’s language the result is immediate, because the rays are gradient lines of \(V\).

Mixed and angle characteristics

Hamilton saw that one can trade positions for directions by a Legendre transformation. For instruments, the most useful version uses the directions of the incoming and outgoing rays as variables (the angle characteristic). Lens designers still use these functions to compute aberrations. The same trick, applied to mechanics, gives the passage from Lagrangian to Hamiltonian variables in Lesson 06.

Conical refraction

For the general reader. Some crystals split a ray of light in two. Hamilton studied the geometry of light in such crystals and found that along certain special directions a single ray should open into a hollow cone. Nobody had seen this. Within two months his colleague Humphrey Lloyd saw it, and the prediction made Hamilton famous across Europe.

Fresnel’s theory

In a transparent crystal the electric displacement \(\mathbf D\) and the electric field \(\mathbf E\) are related by \(\mathbf D=\varepsilon\mathbf E\), with \(\varepsilon\) a symmetric positive matrix. In its principal axes \(\varepsilon=\mathrm{diag}(n_1^2,n_2^2,n_3^2)\) (units with \(\varepsilon_0=1\)). Write \(a=1/n_1>b=1/n_2>c=1/n_3\) for the principal wave speeds, in units of the speed of light. A biaxial crystal has all three different. For a plane wave with unit normal \(\mathbf s\), Maxwell’s equations, or Augustin-Jean Fresnel’s 1821 elastic theory before them, force \(\mathbf D\perp\mathbf s\) and give two possible phase speeds \(v\), the roots of Fresnel’s equation

The energy of the wave travels along the ray direction \(\mathbf t\), parallel to \(\mathbf E\times\mathbf H\), which in general differs from \(\mathbf s\). Plotting the ray speed in every direction gives Fresnel’s wave surface, a quartic surface with two sheets.

Optic axes and the singular points

The two roots coincide, with \(v=b\), exactly when \(s_2=0\) and

These are the optic axes (of wave normals). For such \(\mathbf s\) every direction of \(\mathbf D\) perpendicular to \(\mathbf s\) is allowed, and each gives a different ray. Hamilton realized that the wave surface has four conical points where its two sheets touch, and also four circles along which a plane touches it. Each singular feature means one direction in, a whole cone out.

Derivation sketch

For a given \(\mathbf D\perp\mathbf s\), \(\mathbf E=\varepsilon^{-1}\mathbf D\). Since \(\mathbf H\) is perpendicular to both \(\mathbf s\) and \(\mathbf D\), the ray \(\mathbf t\propto\mathbf E\times\mathbf H\) is perpendicular to \(\mathbf E\) and lies in the plane of \(\mathbf s\) and \(\mathbf D\). It is tilted from \(\mathbf s\) toward \(\mathbf D\) by the angle between \(\mathbf E\) and \(\mathbf D\). Take an orthonormal pair \(\mathbf e_1=(0,1,0)\) and \(\mathbf e_2=\mathbf s\times\mathbf e_1\), and let \(\mathbf D=\cos\theta\,\mathbf e_1+\sin\theta\,\mathbf e_2\). When \(\theta=0\), \(\mathbf D\) lies along a principal axis, \(\mathbf E\parallel\mathbf D\), and the ray coincides with \(\mathbf s\). A short computation shows the tilt is largest at \(\theta=\pi/2\), giving the formula above, and that the transverse displacement of \(\mathbf t\) traces a circle as \(\theta\) runs over \([0,\pi)\). The doubling between \(\theta\) and the angle round the circle is the half-turn property. Michael Berry later identified it as an early instance of a geometric phase.

The experiment

Lloyd used a crystal of aragonite, for which \(\chi\) is about two degrees. On 14 December 1832 he observed external conical refraction, and soon afterward the internal form. A narrow beam sent along the optic axis through a slab emerged as a bright ring. Lloyd also found that the light around the ring was polarized in the way Hamilton’s theory required. James MacCullagh, another Trinity mathematician, argued that the result was implicit in his own earlier work. The dispute soured relations between the two men.

A general method in dynamics

For the general reader. Having described light by one function, Hamilton tried the same with planets and particles. He found that the motion of any mechanical system is governed by a single function of its starting and ending positions. Along the way he rewrote Newton’s laws as a pair of beautifully symmetric equations. These Hamilton’s equations are the form in which mechanics passed into quantum theory.

Hamilton’s principle

Joseph-Louis Lagrange had written mechanics in generalized coordinates \(q=(q_1,\dots,q_N)\) with a Lagrangian \(L(q,\dot q,t)=T-V\). In the 1834 essay, Hamilton stated what is now called Hamilton’s principle: the actual motion between \(q(t_0)=q_0\) and \(q(t_1)=q_1\) makes the action \(\int_{t_0}^{t_1}L\,dt\) stationary. Stationarity is equivalent to the Euler–Lagrange equations. His real innovation was to treat the value of the action as a function of the endpoints.

The proof is the mechanical copy of Lesson 04: vary the endpoint, use the Euler–Lagrange equations along the path, and only the boundary term \(p\,\delta q\) survives. In the first essay Hamilton worked with the characteristic function \(V=\int 2T\,dt\) at fixed energy, the analogue of optical path length. In the second he introduced \(S\), which he called the principal function, and it proved the more useful of the two.

The canonical equations

Assume the Lagrangian is regular: \(\det\big(\partial^2L/\partial\dot q_i\partial\dot q_j\big)\ne0\). Then \(p=\partial L/\partial\dot q\) can be solved for \(\dot q\), and the Legendre transform \(H(q,p,t)=p\cdot\dot q-L\) is a function of \((q,p,t)\).

Proof. Take the differential of \(H=p\dot q-L\): \(dH=\dot q\,dp+p\,d\dot q-\frac{\partial L}{\partial q}dq-\frac{\partial L}{\partial\dot q}d\dot q-\frac{\partial L}{\partial t}dt\). The \(d\dot q\) terms cancel by the definition of \(p\), so \(\partial H/\partial p=\dot q\), \(\partial H/\partial q=-\partial L/\partial q\), \(\partial H/\partial t=-\partial L/\partial t\). The Euler–Lagrange equation \(\dot p=\partial L/\partial q\) becomes \(\dot p=-\partial H/\partial q\). Finally \(\dot H=H_q\dot q+H_p\dot p+H_t=H_t\).

The equations are first order in \(2N\) variables, symmetric between \(q\) and \(p\) up to a sign. For any function \(f(q,p)\), \(\dot f=\{f,H\}\) with the Poisson bracket \(\{f,g\}=\sum_i\big(\partial_{q_i}f\,\partial_{p_i}g-\partial_{p_i}f\,\partial_{q_i}g\big)\). Liouville’s theorem, symplectic geometry and canonical quantization \(\{\,,\}\to\frac{1}{i\hbar}[\,,]\) all start here.

The Hamilton–Jacobi equation and the optical analogy

For the general reader. Hamilton’s principal function obeys one partial differential equation. Carl Gustav Jacob Jacobi saw that solving it, by any trick at all, solves the mechanical problem. The same equation is the eikonal equation of optics in disguise. A century later Erwin Schrödinger used this analogy between rays and trajectories to find wave mechanics.

The equation

Combining \(\partial S/\partial t=-H\) with \(\partial S/\partial q=p\) gives Hamilton’s partial differential equation for \(S(q,t)\), with the initial point held fixed:

Hamilton derived it, together with the equation at the starting point, but used it mainly to characterize his particular \(S\). In 1837 Jacobi showed that any sufficiently general solution will do.

Proof idea. Differentiate \(\partial S/\partial\alpha_j=\beta_j\) along the curve: \(\frac{\partial^2S}{\partial\alpha_j\partial t}+\sum_i\frac{\partial^2S}{\partial\alpha_j\partial q_i}\dot q_i=0\). Differentiating the Hamilton–Jacobi equation in \(\alpha_j\) gives \(\frac{\partial^2S}{\partial t\partial\alpha_j}+\sum_i\frac{\partial H}{\partial p_i}\frac{\partial^2S}{\partial q_i\partial\alpha_j}=0\). The non-degeneracy condition then forces \(\dot q_i=\partial H/\partial p_i\). A similar computation gives the equation for \(\dot p\). In modern terms \(S\) is the generating function of a canonical transformation to coordinates \((\beta,\alpha)\) in which the Hamiltonian is zero.

Rays and particles

For a particle of energy \(E\) in a potential, \(H=|\mathbf p|^2/2m+U(\mathbf x)\). Separating \(S=W(\mathbf x)-Et\) gives

This is the eikonal equation \(|\nabla V|^2=n^2\) of Lesson 04, with refractive index \(n\propto\sqrt{E-U}\). Trajectories are the rays of this fictitious medium, and the surfaces \(W=\text{const}\) are its wavefronts. Hamilton stressed the analogy throughout the 1830s. The analogy is exact only in the ray limit. Schrödinger’s 1926 papers asked which wave equation has the Hamilton–Jacobi equation as its short-wavelength limit, as wave optics has ray optics. Writing \(\psi=e^{iS/\hbar}\) in his equation and letting \(\hbar\to0\) recovers the Hamilton–Jacobi equation.

From couples to quaternions

For the general reader. Complex numbers let you multiply points in a plane. For ten years Hamilton tried to do the same in three-dimensional space and failed. On 16 October 1843, walking to a meeting in Dublin, he saw that it works in four dimensions if you give up one rule of arithmetic: the order of multiplication matters. He carved the rule into a bridge.

Algebraic couples

Complex numbers had been drawn as points of a plane by Caspar Wessel, Jean-Robert Argand and Carl Friedrich Gauss, but their logical status was unclear. In his paper read on 4 November 1833 and published in 1837, Hamilton defined a complex number as an ordered pair of reals with

Then \(i=(0,1)\) satisfies \(i^2=(-1,0)\), and no mysterious square root is needed. This is the construction used in textbooks today. The same paper offered a philosophy, inspired by Kant, of algebra as the “science of pure time”, in which reals are steps in time. The philosophy has not lasted; the construction has.

The failure of triplets

Hamilton wanted numbers \(x+iy+jz\) that could be multiplied with the length of a product equal to the product of lengths, as for complex numbers. In the famous letter to John T. Graves of 17 October 1843 he retold his train of thought. Squaring a triplet gives

The first three parts already have the right length (Exercise 8.1). So the last term must vanish, and he concluded \(ij=-ji\). But then \(ij\) cannot be a combination of \(1,i,j\) (Exercise 8.2). It has to be a new unit \(k\). Space needs four dimensions of number.

Quaternions

The quaternions \(\mathbb H\) are the real vector space with basis \(1,i,j,k\) and the multiplication determined by

extended bilinearly. These relations give \(ij=k=-ji,\ jk=i=-kj,\ ki=j=-ik\). Write \(q=a+\mathbf u\) with scalar part \(a\in\mathbb R\) and vector part \(\mathbf u\in\mathbb R^3\) (Hamilton coined both words). Then

Associativity follows from the identification of \(\mathbb H\) with the real \(4\times4\) matrices of left multiplication (or with the complex matrices \(\begin{pmatrix}\alpha&\beta\\-\bar\beta&\bar\alpha\end{pmatrix}\)). Multiplicativity of the norm, written out in coordinates, is Leonhard Euler’s four-square identity. Two later theorems explain why Hamilton had to go to four dimensions. Ferdinand Georg Frobenius proved in 1877 that \(\mathbb R,\mathbb C,\mathbb H\) are the only finite-dimensional associative division algebras over \(\mathbb R\). Adolf Hurwitz proved in 1898 that real normed composition algebras with unit exist only in dimensions 1, 2, 4 and 8. The eight-dimensional octonions, found by Graves in December 1843 and published independently by Arthur Cayley in 1845, are not associative.

Quaternions, rotations and vectors

For the general reader. Quaternions turned out to be the natural language for rotations in space. A rotation by any angle about any axis is a single unit quaternion, and composing rotations is multiplying quaternions. Spacecraft, robots and video games use them today. Meanwhile the words Hamilton invented, scalar and vector, and the dot and cross products hidden in his product, became the vector analysis of physics.

Proof sketch

Conjugation preserves the norm and the scalar part, so it maps pure quaternions to pure quaternions isometrically, and it is linear. Since \(S^3\) is connected and contains 1, the image lies in \(\mathrm{SO}(3)\). It fixes \(\mathbf n\), because \(q\) commutes with \(\mathbf n\). For \(\mathbf v\perp\mathbf n\), the product formula gives \(\mathbf n\mathbf v=\mathbf n\times\mathbf v\) and \(\mathbf v\mathbf n=-\mathbf n\times\mathbf v\). Write \(c=\cos\frac\theta2,\ s=\sin\frac\theta2\) and use \(q^{-1}=\bar q\):

using \(\mathbf n\mathbf v\mathbf n=(\mathbf n\times\mathbf v)\mathbf n=-(\mathbf n\times\mathbf v)\times\mathbf n=\mathbf v\) when \(\mathbf v\perp\mathbf n\), \(|\mathbf n|=1\). This is rotation by \(\theta\) in the plane perpendicular to \(\mathbf n\). Every rotation has an axis and an angle, so the map is onto. The kernel consists of the quaternions commuting with every pure quaternion, the reals, of which the unit ones are \(\pm1\). In modern language \(S^3=\mathrm{SU}(2)\) is the double cover of \(\mathrm{SO}(3)\), the group behind electron spin.

Vector analysis

For two pure quaternions, \(\mathbf u\mathbf v=-\mathbf u\cdot\mathbf v+\mathbf u\times\mathbf v\). Hamilton introduced the operator \(\nabla=i\,\partial_x+j\,\partial_y+k\,\partial_z\). Applied to a vector field it gives \(\nabla\mathbf F=-\operatorname{div}\mathbf F+\operatorname{curl}\mathbf F\), and \(\nabla^2=-\Delta\). Peter Guthrie Tait in Edinburgh became the champion of quaternions, and James Clerk Maxwell used their notation alongside Cartesian components in his 1873 Treatise on Electricity and Magnetism. In the 1880s Josiah Willard Gibbs and Oliver Heaviside split the product into the separate dot and cross products, and their vector analysis displaced quaternions in physics after a bitter dispute with Tait. Quaternions returned in the twentieth century through spin, Clifford algebras, and computer graphics.

In the Lectures of 1853 Hamilton also showed that a linear function of quaternions satisfies a polynomial equation built from its own invariants, a special case of what Cayley stated for matrices in 1858. The general result is the Cayley–Hamilton theorem.

The icosian calculus

For the general reader. In 1856 Hamilton invented another non-commutative algebra, with three symbols whose powers return to 1. It describes the symmetries of the icosahedron. He turned it into a puzzle, the icosian game, which asks for a round trip along the edges of a dodecahedron visiting every corner once. Such round trips are now called Hamiltonian cycles, and finding them in general graphs is one of the hard problems of computer science.

The relations

In a memorandum of 1856 Hamilton introduced symbols \(\iota,\kappa,\lambda\) with

Hamilton read these symbols as operations on the directed edges of a dodecahedron. \(\iota\) reverses an edge. \(\kappa\) turns an edge about its initial vertex to the next edge there. Then \(\lambda=\iota\kappa\) and \(\mu=\iota\kappa^2\) advance along a path, turning one way or the other at the next vertex. A word in \(\lambda,\mu\) is a route. Hamilton found that the word \((\lambda^3\mu^3\lambda\mu\lambda\mu)^2\) equals 1 and that the corresponding route passes through all twenty vertices before returning: a Hamiltonian cycle. In the hero picture the turns are written R and L.

The game

The board is a flat drawing of the dodecahedron, the graph above, with twenty holes for numbered pegs. Hamilton showed it at the 1857 British Association meeting in Dublin. In 1859 he sold the rights to the London games firm Jaques for £25. It did not sell well, perhaps because it is too easy once you know the trick. Thomas Kirkman had independently studied cycles through all vertices of polyhedra at about the same time, and Hamilton sent him a copy of the game. Hamilton’s name stuck because of the game.

The quaternion years, 1843–1865

In 1837 Hamilton was elected President of the Royal Irish Academy, and on 16 October 1843 he was walking with his wife along the Royal Canal to preside at a meeting of its Council. As they reached Broome Bridge (he wrote Brougham), the relations \(i^2=j^2=k^2=ijk=-1\) came to him, and he cut them into the stone with a knife. The carving has long gone. A plaque was unveiled at the bridge in 1958, and mathematicians now walk the route each 16 October. He presented the discovery to the Academy on 13 November 1843.

He spent the rest of his life on quaternions. He believed they would be as important for physics as calculus had been, and he applied them to geometry, optics and dynamics. The Lectures on Quaternions (1853), more than seven hundred pages long, proved hard to read; even admirers found it impenetrable. He then began a textbook, Elements of Quaternions, which grew far beyond its plan.

Private troubles

His later life was lonely. His wife was often away, and his study at Dunsink filled with papers. In 1846 he was embarrassed by drunkenness at a scientific dinner in Dublin and gave up alcohol for two years, but then returned to it. In 1848 Catherine Disney, the love of his youth, now unhappily married, began to correspond with him. The correspondence brought him distress and grief. She died in 1853, about two weeks after he gave her a copy of the Lectures. He continued to write poetry and to correspond with friends such as Augustus De Morgan, with whom he exchanged long letters for years.

Recognition kept arriving. The Royal Irish Academy gave him its Cunningham Medal for the second time in 1848. In 1864 the newly founded National Academy of Sciences in the United States elected its first Foreign Associates and placed his name at the head of the list. News reached him during his final illness.

Death and legacy

Hamilton suffered a severe attack of gout in 1865 and died on 2 September 1865 in Dublin, aged 60. He was buried in Mount Jerome Cemetery. Elements of Quaternions was almost finished. His son William Edwin Hamilton completed the edition, and it appeared in 1866, about 760 pages long. Robert Perceval Graves, a friend, published a three-volume Life between 1882 and 1889.

Legacy

• Mechanics: the Hamiltonian, Hamilton’s equations and the Hamilton–Jacobi equation are the backbone of classical mechanics, celestial mechanics, statistical mechanics and symplectic geometry. Through canonical quantization, the Hamiltonian became the central operator of quantum mechanics.
• Optics: the characteristic function underlies aberration theory and the eikonal method. Conical refraction is still studied, for example by Michael Berry and coworkers, and is used in optical trapping and laser beam shaping.
• Algebra: the definition of complex numbers as pairs; the first non-commutative division algebra, which opened the way to Cayley’s matrices, Grassmann’s and Clifford’s algebras and modern abstract algebra; the vocabulary of scalars and vectors.
• Graph theory: Hamiltonian cycles and paths, and the NP-complete problem of deciding whether they exist.
• Honours: Knighthood and Royal Medal (1835), two Cunningham Medals (1834, 1848), first Foreign Associate of the US National Academy of Sciences (1864). The Hamilton Mathematics Institute at Trinity College Dublin bears his name, and the lunar crater Hamilton is named for him.

Connections

People linked to William Rowan Hamilton across this collection. A name without a link has no page here yet. See William Rowan Hamilton in the network of all 337 lives →

Influenced by

• Alexis Clairaut
• Augustin-Jean Fresnel
• Isaac Newton
• Joseph-Louis Lagrange
• Pierre-Louis Moreau de Maupertuis
• Pierre-Simon Laplace
• Siméon Denis Poisson
• Étienne-Louis Malus
• John Brinkley

Influenced

• Carl Gustav Jacob Jacobi
• Erwin Schrödinger
• James Clerk Maxwell
• Josiah Willard Gibbs
• Peter Guthrie Tait

Colleagues and collaborators

• Arthur Cayley
• Hermann Grassmann
• János Bolyai
• Humphrey Lloyd 1832
• Thomas Kirkman

Rivals

• James MacCullagh

Friends

• Arthur Cayley
• Augustus De Morgan
• John T. Graves 1843
• William Wordsworth 1827-1850

Also mentioned in this life

• Leonhard Euler
• Pierre de Fermat
• Georg Frobenius
• Carl Friedrich Gauss
• Oliver Heaviside
• Adolf Hurwitz
• Adrien-Marie Legendre
• Joseph Liouville

Fields

• optics 38 lives
• algebra 29 lives

On CoursesHub

• Lagrange, Hamilton & Variational Principles · History of Mathematics & Physics
• Optics & Photonics
• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Sir William Rowan Hamilton, MacTutor History of Mathematics.
2. William Rowan Hamilton, Wikipedia (with extensive references).
3. D. R. Wilkins, Sir William Rowan Hamilton (1805–1865), Trinity College Dublin: transcriptions of the major papers and letters.
4. T. L. Hankins, Sir William Rowan Hamilton, Johns Hopkins University Press, 1980.
5. R. P. Graves, Life of Sir William Rowan Hamilton, 3 vols., Dublin, 1882–1889.
6. Royal Irish Academy, history on MacTutor.

Mathematics and physics

1. W. R. Hamilton, Theory of Systems of Rays, Trans. Roy. Irish Acad. 15 (1828), and the third supplement, Trans. Roy. Irish Acad. 17 (1837).
2. Conical refraction: Hamilton’s and Lloyd’s papers, Trinity College Dublin.
3. W. R. Hamilton, On a General Method in Dynamics, Phil. Trans. R. Soc. (1834); Second Essay on a General Method in Dynamics, Phil. Trans. R. Soc. (1835).
4. W. R. Hamilton, Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, Trans. Roy. Irish Acad. 17 (1837).
5. W. R. Hamilton, Letter to John T. Graves on quaternions, 17 October 1843; Hamilton’s research on quaternions.
6. W. R. Hamilton, Lectures on Quaternions, Dublin, 1853; Elements of Quaternions, London, 1866.
7. Conical refraction and icosian calculus, Wikipedia.
8. M. V. Berry and M. R. Jeffrey, Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics, Progress in Optics 50 (2007).
9. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 2nd ed. 1989.
10. J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, 2003.
11. M. J. Crowe, A History of Vector Analysis, University of Notre Dame Press, 1967.