Sir William Rowan Hamilton did not make one major discovery but two; a way to describe mechanics which is now called Hamiltonian mechanics, and the quaternions, an extension to three dimensions of the imaginary numbers.
In the many theories of modern physics, the evolution of a physical system is largely controlled by its Hamiltonian, describing the total energy of that system. In mathematics, the Hamiltonians play a major role in dynamical systems, differential equations, Lie group theory, and symplectic geometry.*
Although in the 1880s vector analysis was directly derived from the quaternions, in physics the quaternions were thereafter forgotten for a long time because vector analysis was far more intuitive and easy to work with. But the quaternions again emerged with the coming of computers, which can more easily work with them than with vector analysis. The quaternions were used in the Space Shuttle program, found their way into games and films, and are now widely used for smoothly changing orientations in for instance robotics, gaming and spacecraft, from your phone up to the rovers on Mars.
The Dublin and Kingstown Railway was Ireland's first railway,
the locomotive depicted here was built in 1834.
Copied from the Dublin Penny Journal of September 1834.
This engraving, of the steam locomotive Hibernia, shows the state of science a year after Hamilton’s marriage, two years after the publication of the Third Supplement to an Essay on the Theory of Systems of Rays,** which would lead to Hamiltonian mechanics, and nine years before Hamilton found the quaternions.
Thus having made his discoveries in a time in which there were no telephones and radios yet, people travelled by horse, carriage, steam train and steamship, Hamilton’s son reported that “Sir W.” was indifferent to contemporary fame, “arising from his conviction that his belonged to a future age entirely.” He was right, yet little could he know that his work had to wait until quantum mechancis and the dawn of computers before it would start to blossom.
His work being very widely acknowledged now, it is time to brush up his reputation and show him as the happily married genius he was.
* After Terence Tao, Hamiltonians. Tao continues, “Roughly speaking, each physical phenomenon (e.g. electromagnetism, atomic bonding, particles in a potential well, etc.) may correspond to a single Hamiltonian H, while each type of mechanics (classical, quantum, statistical, etc.) corresponds to a different way of using that Hamiltonian to describe a physical system.”
** For short descriptions of the original essay and the three supplements see David Wilkins’ Theory of Systems of Rays.