If ever there was a child prodigy, Hamilton was it. He grew up with his uncle who was a bit of an eccentric; for instance, he tied a string around young William's toe at night, ran it through a hole in the wall into his own bedroom, and then early each morning he would tug on the string to wake him and start him on his studies. By the age of 12, William was fluent in 10 languages and was appointed to a Mathematics Chair at the Royal Observatory in Dublin at a youthful age. One of Hamilton's successes was in proving that Newton's Equations and Lagrangian Mechanics were equivalent when the Lagrangian was the difference between the kinetic and potential enrgy of a system. Up to that time there were arguments over which was correct. He showed that they were different manifestations of the same thing.
He also put some ideas of Fermat into a new perspective. We mentioned that Fermat came to understand the propagation of light based on the idea that it would follow a path of least time. Hamilton took Lagrange's action property and showed that the path a particle would take would be the path of least action. This correlation between the motion of waves (as light was undertood at the time) and that of particles could be described in a common way, hints, in retrospect at least, to the wave-particle duality that we understand in modern quantum theory. If experiments could have been up to the task at the time, it is very likely that Hamilton would have developed quantum theory one century before Schröodinger.
Perhaps Hamilton's most important contribution came from his reformulation of Newton's Laws. In the same way that Lagrange provided a new method for solving mechanical problems, Hamilton put forward still another formalism. He showed that the results were equivalent in the three methods, but his proves to be most useful for a certain class of problems. Even today, Hamiltonian Mechanics is used commerically to determine orbital trajectories of satellites.
The theory stems from a new variable called the Hamiltonian, which is the sum of the system's kinetic and potential energy. And the equations of motion derive from this. It turns out that Hamiltonian mechanics were the starting point for Schrödinger's development of his Wave Mechanics, the classical theory simply twisted to account for the quantum observations.