In 1823, he enrolled at Trinity College in Dublin, where he earned high honors in mathematics and classics. After several years of research in the field of optics, Hamilton submitted a paper on the "Theory of Systems of Rays" to the Royal Irish Academy. His paper revolutionized the field of geometrical optics by using Fermat's principle that light follows the path of least time. Hamilton borrowed from Lagrange's action property to prove that a particle follows the path of least action.

Hamilton's paper eventually led to the post of Andrews Professor of Astronomy and Royal Astronomer of Ireland (he was only 22 at the time). In 1835, he was the chief local organizer of the British Association for the Advancement of Science meeting in Dublin. At the closing dinner of the meeting, he was knighted by the lord lieutenant and became Sir William Hamilton. About two years later, he became president of the Royal Irish Academy. In 1843, he was awarded a life pension by the British government.

Hamilton made several contributions to mathematics and physics. He discovered the phenomenon of conical refraction. He reformulated Newton's Laws into a form used today to determine orbital trajectories of satellites. His theory of quaternions, a complex form of calculus, developed into modern abstract algebra. His unification of optics and dynamics greatly affected mathematical physics about a century before the introduction of quantum theory. Some have even speculated that Hamilton could have introduced quantum theory before Schrodinger, because Hamiltonian Mechanics were the starting point for Schrodinger's development of wave mechanics.