Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. A mathematical function that can be used to generate the equations of motion of a dynamic system, equal for many such systems to the sum of the kinetic and potential energies of the system expressed in terms of the system's coordinates and momenta treated as independent variables.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Applied to various mathematical constructs developed or inspired by Hamilton, as in Hamiltonian path, Hamiltonian cycle.
 n. In quantum mechanics, the observable, denoted by H, that corresponds to the total energy of the system.
 n. A native or inhabitant of any city named Hamilton.
from The Century Dictionary and Cyclopedia
 Pertaining to James Hamilton (1769–1831), and especially to a system of teaching languages which he advocated, and which was based upon the two principles that language is to be presented to the scholar as a living organism, and that its laws are to be learned by observation and not by rules.
 Pertaining to Sir William Hamilton (1788–1856), an influential philosopher and logician of the Scottish school.
 Pertaining to Sir William Rowan Hamilton (1805–65), an Irish mathematician.
 Pertaining to or holding the political doctrines of Alexander Hamilton (1757–1804), an American statesman, who was one of the leaders of the Federalist party and the first Secretary of the Treasury.
 n. A follower of any one of the persons named above. See I.
Etymologies
Examples

I randomly opened a cupboard today to find some interesting stuff – an old journal paper of Dad’s (Hi Dad!) on Liapunov equations in Hamiltonian systems and a diary from when he was around 15.

Agreeing with Aaron: once we represent the results of experiments by operators acting on a Hilbert space, time evolution gives a oneparameter group, so by Stone’s theorem there is a selfadjoint operator, which in the case of time evolution we call the Hamiltonian, that generates the group.
Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask

(If a Hamiltonian is a sum of polynomially many local terms, then certainly one can simulate it efficiently on a quantum computer — see here for some pointers to the literature.)

For the interaction picture one splits up the Hamiltonian, which is the generator of timetranslations, into two parts H = H0

Because the oscillators are independent, the Hamiltonian is a simple sum: we see that the Hamiltonian of the EM field can be looked upon as a Hamiltonian of independent oscillators of energy ω = 

It smacks of the same kind of Hamiltonian authoritarianism of the global warmers and environmental extremists.

However, chaotic systems such as Hamiltonian chaotic systems will exhibit a radical divergence of these errors or differences in initial conditions

PAUL STREET  Obama was recently hailed as a "Hamiltonian" believer in "limited government" and "free trade" by Republican New York Times columnist David Brooks, who praises Obama for having "a mentality formed by globalization, not the SDS." ...

PaleoProgressives were not, and are not "New Deal Democrats,"  an appellation that has appropriately been coopted and appropriated by the selfdescribed "Hamiltonian" LaRouchies.

1354 R.A. A.BERTY tion of coordinates and velocities, but it is often more convenient to define the Hamiltonian function H with a Legendre transform because the Hamiltonian is a function of coordinates and momenta.
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