Tuesday, August 04, 2015

William Rowan Hamilton: 210th birthday

Sir William Rowan Hamilton (1805-1865) was born on midnight between 3rd and 4th August, 1805, to Sarah Hutton and Archibald Hamilton (a solicitor) in Dublin. The couple had 9 children. Statistics shows that in such multiplets with 9 components, the 4th oldest child has the highest chance to become a mathematical genius. And it was Hamilton's case, too.

(Let me admit that the statistical conclusion largely depends on one entry, Hamilton's family itself LOL.)

As a kid, he had to learn the simple things that almost everyone can learn. By his 7th birthday, he has mostly mastered Hebrew. And before his 13th birthday, he added about 6 European languages plus Persian, Arabic, Hindustani, Sanskrit, Marathi, and Malay – in the last two cases, I didn't even actively know the language's existence but the kid spoke them. His uncle, a linguist, was helpful for William to accumulate that knowledge. ;-)

As a teenager, he realized that linguistics and languages are too primitive things, so he downgraded them to a hobby. Throughout his life, he would read what's shaking in Persian or Arabic in his spare time. But he began to do some serious physics and mathematics. He was always sure about these disciplines' superior importance – a fact obvious from his quotes:
Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?

On earth there is nothing great but man; in man there is nothing great but mind.

When he was 18, before he made any advance we remember, astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton: "This dude, I do not say will be, but is, the first mathematician of his age." And the history made the bishop right – even though we could argue that it was done retroactively.

When Hamilton was between 19 and 24, he was preparing a "paper" for the Irish Royal Society. It was on "caustics". The referees insisted he would make it clearer so he made it more readable – but also longer – and as a result, he described the formalism to unify optics and mechanics.

He figured out that Hamilton's principal function – the player in Hamilton's principle – may be placed at the center of all dynamics. Yes, this is nothing else than the action \(S\) that we associate with the competing, "Lagrangian" approach to classical mechanics.

The modern terminology that describes the Lagrangian and Hamiltonian approaches to classical physics as two "opponents" is a ludicrous distortion of the history because both of them were pretty much settled by Hamilton himself.

Joseph Lagrange lived some 70 years earlier and he realized that "something" had to be minimized and one can derive the right Newton-like equations – the Euler-Lagrange equations – from that condition. But he wasn't really able to say what was the functional that we minimized in complete generality. So all the universal formulations of all versions of advanced classical mechanics were only settled by William Rowan Hamilton when he was in his 20s. (Jacobi added some "even more rigorous" insights after Hamilton which is why we often talk about the Hamilton-Jacobi equations.)

I often emphasize that quantum mechanics is more fundamental and primary. It was only explicitly formulated in the 1920s, exactly 100 years after Hamilton's big insights on mechanics (and optics). But there were "trends" that were directing physics towards quantum mechanics long before the 1920s.

Needless to say, the experimental discoveries of the stable atom, finite energy emitted by the black bodies, radioactivity, and lots of other things were "predicting" the rise of a new theory. That's the trivial part of the story. However, even at the theoretical level, quantum mechanics was getting ready to arrive.

In thermodynamics and statistical physics, Ludwig Boltzmann figured out that the thermodynamic notions of entropy and temperature had a microscopic explanation. The entropy was \(S=k\ln W\) where \(W\) was the number of possible states. This formula from Boltzmann's tomb is particularly natural if \(W\) is a genuine integer and the phase space of possible states is therefore quantized. That was one reason that would make Boltzmann quickly embrace (or discover) quantum mechanics if he had not committed suicide too early. But there were other reasons: he was deriving deterministic statements about thermal phenomena from probabilistic distributions and probabilistic predictions of statistical physics. Quantum mechanics generalized that approach in a rather straightforward way. But Boltzmann already knew that important truths about Nature have to be derived from intrinsically probabilistic calculations.

But even decades before Boltzmann's realizations, Hamilton found those two functions – the Hamiltonian and the Lagrangian – which become much more important in quantum mechanics (either in the operator approaches or the Feynman path-integral formulation) than they are in classical physics. In classical physics, the equations may be reinterpreted in terms of the Hamiltonian and the Lagrangian which is "cool" but it seems unnecessary to do science.

Even though quantum mechanics was only written down 60 years after Hamilton's death, I think it is obvious that Hamilton's insights should be viewed as important theoretical steps towards quantum mechanics, too. He appreciated "some structure" underlying classical physics whose existence may be explained if classical physics is a limit of a deeper and in some sense simpler theory, quantum mechanics.

In the late 1820s, he inherited the astronomer job from the bishop who had rated the 18-year-old guy to be the #1 mathematician in the world. Hamilton knew a lot about and contributed things to optics – both at the theoretical and practical level.

On October 16th, 1843, Hamilton was walking along the Royal Canal in Dublin with his wife when the equation\[

i^2 = j^2 = k^2 = ijk = -1

\] appeared in front of his eyes. So he ignored his wife (it's her job not to fall to the canal, after all), took his penknife, and carved the equation generalizing the complex numbers into the nearby Brougham Bridge (which the non-Hamiltons, non-Motls call the Broom Bridge). That's how the quaternions were discovered – even though all the algebraic structure (without a catchy name) had been independently known to Spanish banker Benjamin Olinde Rodrigues since 1840 (yes, it is truly ironic that bad marketing became the main reason why this banker isn't credited with the birth of quaternions).

Because the bridge seems to have such a great track record in creation of great mathematical ideas, mathematicians and physicists are visiting it every year for a ritual walk. Attendees have included Weinberg, Gell-Mann, Wilczek; Wiles, Penrose, Daubechies. These guys are walking and walking and walking but whether something aside from quaternions has emerged out of it isn't clear. My heretical, politically incorrect idea could be that the discovery of the quaternions could have something to do with Hamilton and not just the bridge. ;-)

He also coined some generalizations – for example, "biquaternions" are quaternions whose four real components are complexified. I don't think it's particularly natural or sensible but OK. Similar comments apply to the icosian calculus (he also coined) which boast "new" imaginary units that are 5th roots of unity and are demanded not to commute with each other etc. In those objects, the non-Abelian finite group should really be separated from the real numbers parameterizing the components.

Quaternions are beautiful but they were overrated. Since the 1840s, Hamilton and others would try to rewrite many things (e.g. Maxwell's equations in their original form) in terms of quaternions which seems like misguided steps to me because the relevant symmetries (automorphisms) are often different. In particular, I have a problem with Hamilton's quote:
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.
The unification of space and time is amazing for a 19th century quote – it sounds like Minkowski's 1908 reformulation of special relativity in terms of a spacetime. However, there is a technical bug that Hamilton couldn't have been aware of: in relativity, the relevant group is \(SO(3,1)\) while the automorphism of the quaternions is just \(SO(3)\). For this reason, only the 3-vector part of the physical objects is captured correctly by the quaternions. Both the relativistic spacetime and quaternions add a fourth coordinate but its behavior relatively to the previous three coordinates is different in the spacetime and in the quaternions.

In 1846, Hamilton invented the "hodograph", a method to visualize the motion of an object in a fluid.

Hamilton has also influenced the mathematical notation. He coined the nabla \(\nabla\) symbol, the word "tensor", and – less successfully – the "versor" which is the \(SO(3)\) rotation matrix written as \(\exp(a \cdot {\bf r})\) where \({\bf r}^2=-1\) is a purely imaginary unit quaternion (defining the axis of the rotation). In 1853, Hamilton proved the Cayley-Hamilton theorem: if you substitute the matrix \(A\) itself for \(\lambda\) to its own characteristic polynomial \(\det(\lambda\cdot I - A)\), you get zero.

He has also influenced the "philosophical layer of mathematics". For example, he and deMorgan originated the "universal algebra" which was (or is) a predecessor of the category theory. Some people think that category theory may define the "future of theoretical physics" much like the Hamiltonians and Lagrangians defined the "future of physics" in the middle of the 19th century. Maybe they're right. I don't see it at this point, however.

The man has also dealt with the recreational mathematics – e.g. the icosian game where one finds some path through the vertices of the icosahedron. (The Hamiltonian path is not a trajectory in the sense of Feynman's path integrals but instead a notion in graph theory – a path in which you visit each vertex once.) He was a prolific correspondent. Sometimes, his letter would focus on a relatively narrow problem and he would write 100 pages of text about it to someone. I won't humiliate Hamilton for that because that could be like if I were humiliating myself. ;-)

Hamilton has done very important things and enjoyed his life. He ate a lot of great food (only when he was not thinking hard: at those special times, he denied the existence of earthly urges, as his son noted) and drank some good beverages. A severe gout came in 1865, when he was 60 years and 1 month old, and killed him. Death is usually an unwise career move but for geniuses of Hamilton's caliber, the successes are only getting started at the moment of death.

No comments:

Post a Comment