Hamilton: Mathematician and Romantic - Prof. Luke Drury, President of Royal Irish Academy. For additional information including audio recordings to accompany this presentation please click here - http://www.ria.ie/library/exhibitions/lunchtime-lecture-series.aspx. Disclaimer: The Royal Irish Academy has prepared the content of this website responsibly and carefully, but disclaims all warranties, express or implied, as to the accuracy of the information contained in any of the materials. The views expressed are the authors’ own and not those of the Royal Irish Academy.

1. Hamilton
Mathematician and Romantic
Luke Drury

2. Why Hamilton?
Hamilton is to Irish Science what Joyce is to
Irish literature - the preeminent figure whose
global reputation transcends his Irishness.
2005, the bicentenary of his
birth, was declared to be the
“Hamilton Year - celebrating
Irish Science”

3. Hamilton with
one of his sons
(circa 1845)

4. Hamilton presiding as president of the RIA
(from the Dublin University Magazine, June 1842)

5. Hamilton in middle age

6. Hamilton at age 54
(after Graves)

7. Two stamps issued in 1943 (in the middle
of the “emergency”!) to commemorate
the discovery of quaternions.

8. Rather better in 2005!

9. Brief chronology
Born 3-4 August 1805 in Dominick Street
Father was Archibald Hamilton, lawyer and
estate agent for the patriot Archibald
Hamilton Rowan.
Mother was Sarah Hutton, from a family of
Dublin coachbuilders.
One Scottish, One French and two Irish
grandparents.

10. Sent at age 3 to his Uncle James
Hamilton, curate of Trim and master of
the Meath diocesan school housed in
Talbot’s castle (now marked with a
commemorative plaque).
Major educational influence - Uncle
James was a graduate of TCD with a
strong interest in education and
languages who believed in starting as
early as possible with a very broadly
based education and lots of hard work.

12. The myth of 13 languages by 13
WRH certainly was remarkably fluent in
Latin, Greek and Hebrew from a very
early age.
He appears to have had a basic
knowledge of Persian, Arabic, Syriac and
some Sanskrit
Impressive, but hardly 13. (NB no Irish!)
In later life could read, but not speak,
French, German and probably Italian.

13. Remarkably, little evidence of early
mathematical abilities!
Dec 14, 1815 to his sister Grace.
I have for some time been reading Lucian and Terence, the Hebrew
psalter on Sundays and on Saturdays some Sanskrit, Arabic and Persian.
I read at leisure hours Goldsmith’s Animated Nature and any new
history or poetry than falls my way. I like Walter Scott very much. In
arithmetic I have got as far as practice and I have done very near half
the first book of Euclid with my uncle.
I do the ancient and modern geography of the different countries
together. I do the second lesson every morning in the Greek
Testament and, on Sundays after church go over the scripture lessons
of the past week with Dodderidge’s Notes and Improvements, and
before church I read Secker on the Catechism, and in the evenings
Wells’ Scripture Geography, a very entertaining book.
Extract from earliest surviving letter of WHR

14. Mother dies in 1817 (when he is 12).
Father dies in 1819
Meets Zerah Colburn, 1818 and 1820.
Develops increasing interest in
mathematics
1823 - comes first in TCD entrance
examination with special prize in Hebrew.

15. Wins virtually all prizes in TCD
Aug 17th 1824, meets Catherine Disney
May 5th 1825, Catherine is married to Rev
Barlow
April 27 1827, submits paper on Systems of
Rays to PRIA
June 16th 1827, appointed to Dunsink
Sep 15th 1827, first meeting with Wordsworth

17. Aug 1829, Wordsworth visits Dunsink
March 1832, visits Coleridge in London
22 Oct 1832 Predicts conical refraction
April 9 1833, marries Helen Bayly
April 10 1834, submits paper on General
Methods in Dynamics

18. June 1 1835, presents ideas on Algebra as
Science of Pure Time
Aug 15 1835, knighted at BAAS meeting
Dec 1837, elected president of RIA
1840 Helen leaves for two years
Oct 16 1843, discovers quaternions

19. Feb 11 1846, incapacitated at meeting of
the Geological Society - abstains for two
years.
Oct 1853, Catherine Disney/Barlow dies
Jan 9 1865, elected first foreign associate
of the NAS
Sep 2 1865, dies at Dunsink

20. Hamilton’s Optics
First part of system of rays deals with
rays reflected from arbitrary mirrors.
Extended to arbitrary refracting media in
part two.
Finally to anisotropic crystalline media in
third supplement.
Based on Fermat’s principle of least time

21. The actual path followed by light travelling from
X to X’ via a reflection from the mirror surface
is the shortest such path, and thus the light travels
from X to X’ in the shortest time.

22. Similarly for refraction - can show that
light takes path of shortest time
between two points if speed in media is
inversely proportional to the “index of
refraction” n.
Z
δ n ds = 0

23. How lens works - light travels more slowly in glass
than air so all light paths take the same time.

24. Hamilton then introduces his
“Characteristic function” V(x’,x) of a
general optical system as essentially
this minimum (more precisely
stationary) time for light to travel
from x’ to x through the system. All
the properties of the system are
contained in this one function!
Z x
V (x , x) =
0
nds
x0

25. The really important difference in
Hamilton’s approach is that he allows both
end points to vary as well. For isotropic
media this gives the “eikonal equations”
✓ ◆2 ✓ ◆2 ✓ ◆2
∂V ∂V ∂V
+ + =n 2
∂x ∂y ∂z
✓ ◆2 ✓ ◆2 ✓ ◆2
∂V ∂V ∂V
+ + =n 02
∂x0 ∂y0 ∂z0

26. For general anisotropic media the
refractive index depends on the direction
of propagation as well as position,
d xi
n = n(xi, αi), αi =
ds
But the direction cosines of the ray
are not independent quantities...
αiα = 1
i

27. Hamilton however extends the refractive
index to a homogeneous function of first
order in the direction cosines treated as
independent quantities. Then by Euler’s
identity,
∂n
n = αi
∂αi
(Easiest way to think of this is to take a
function defined on the unit sphere and
then scale it proportional to the radius for
points off the unit sphere)

28. Z Z Z
δ n ds= δn ds + n δds
Z ✓ ◆ Z
∂n ∂n ∂n
= δxi + δαi ds + αi δds
∂xi ∂αi ∂αi
Z Z
∂n ∂n
= δxi ds + (δαi ds + αi δds)
Z ∂xi Z ∂αi
∂n ∂n
= δxi ds + δdxi
Z
∂xi  ∂α
✓ i◆  x
∂n d ∂n ∂n
= δxi ds + δxi
∂xi ds ∂αi ∂αi x0

29. This gives Hamilton’s starting point for
his geometrical optics of arbitrary
media;
✓ ◆
d ∂n ∂n
=0
ds ∂αi ∂xi
gives the equation of the ray, and the
variation of the characteristic function
when the end points move is given by
∂V ∂n ∂V ∂n
= , 0=
∂xi ∂αi ∂xi ∂α0i

30. Mechanics also can be expressed as
the principle of “least action”
Z
S= Ldt
δS=0
Thus Hamilton’s ideas carry over to
general mechanics. The equivalent
of the “eikonal” equation is the
Hamilton-Jacobi equation.

31. How (or why) can nature follow the path
of stationary action?? The only way to
know that the action is stationary is to
explore all the paths from an inital to a
final state.
This is precisely what happens in
quantum mechanics!
Z 2
P(A ! B) = eiS

32. QM removes the “unphysical” suspicion of
teleology from the principle of least
action (originally introduced on
theological grounds in 18th century!) but
at the expense of the `spookiness’ so
detested by Einstein and Schroedinger!
Perhaps not inappropriate for
Halloween....

33. Herr Nietsche said
"God is dead!"
Herr Drury said "that's crazy,
my reflection
on 'Least Action'
tells me He's just lazy!"
(Iggy McGovern, personal communication, 2005)

34. Hamilton’s focus on the mathematical
structure of systems defined by
variational principles has remained at the
heart of theoretical physics since his day
and is still there! However precisely
because it is so fundamental and
universal it is not easy to convey to
nonspecialists just why it is so important.

35. Other major contribution was the
discovery of quaternions.
Hamilton “demystified” imaginary (or
complex) numbers by regarding them as
pairs of real numbers with appropriate
rules for addition, multiplication etc.
(a, b) + (c, d)=(a + c, b + d)
(a, b) ⇥ (c, d)=(ac bd, ad + bc)

36. Obvious question is can one extend this
idea to triples of real numbers?
Surprisingly no, but as Hamilton suddenly
realised it will work if one goes from
three to four real numbers! Hence the
quaternions.
But - multiplication is noncommutative!
ab 6= ba

37. A general quaternion has the form
a + bi + c j + dk
with a scalar part a and vector part
(b,c,d). Every nonzero element has an
inverse
a bi c j dk
a 2 + b2 + c2 + d 2

38. Important mainly for introducing the idea
of non-commutative multiplication
(another key idea in quantum mechanics!)
Never achieved the importance Hamilton
thought they would, but beautiful and
useful.
(Only four division algebras exist- the
reals, the complex numbers, quaternions
and octonions)

39. Hamilton also made important
contributions to graph theory
(Hamiltonian circuits), the theory of
fourier series and to Abel’s proof that
the general quintic was not soluble in
radicals.

40. His lasting fame however relies on his
fundamental formulation of optics and
mechanics into what is now called
canonical or Hamiltonian form, a form so
fundamental that it has survived the
transition from classical to quantum
mechanics!

42. Hamilton the romantic.
Highly idealistic he was a deep admirer
of the romantic poets and considered
poetry to be much superior to prose.
Also in his doomed and unhappy love for
Catherine he displayed a typically
romantic streak to his nature.
Was introduced to German idealism by
Coleridge - Kant was a major influence.

43. One reason his works are so hard to read
is that he sought to write a form of
mathematical poetry whereby the
maximum meaning is conveyed with the
minimum content - distilling out the
essence of each problem and presenting
it in the most general and abstract form.
But of course this is also why his
contributions are so fundamental.

44. Surely a lesson for us today - long term
impact comes from a focus on deep
problems and underlying structures, not
on immediate applications. Also we need
to see science as part of culture and as
a creative activity closely allied to the
creative arts.

45. Less Science and Technology, more
Science and Culture!