In Lecture Notes on Quaternions, Hamilton discussed the problem that linear transformation satisfies its characteristic polynomial. In an article in 1858, A.Cayley verified this theorem for the case of n=3, but thought it unnecessary to prove it further. Frobenius (F.G.Frobenius) is in 1858.
Hamilton's Personal Contribution
Hamilton works hard and has an active mind. Published papers are generally concise and difficult for others to understand, but the manuscripts are very detailed, so many achievements are compiled by later generations. There are only 250 notes of Hamilton's manuscript, a large number of academic correspondence and unpublished papers in the library of Trinity College. The National Library of Ireland also has some manuscripts.
His research involves many fields, and his greatest achievements are optics, mechanics and quaternion. The optics he studied is geometric optics, which has mathematical properties; Mechanics is to list dynamic equations and solve them; So Hamilton is mainly a mathematician. But his contribution to mechanics is the most influential in the history of science. Hamilton quantity is the most important quantity in modern physics.
Hamilton developed analytical mechanics. 1834, the famous Hamilton principle was established, so that various dynamic laws can be deduced from a variational formula. According to this principle, there are similarities between mechanics and geometric optics. Later, it was found that this principle can be extended to many fields of physics, such as electromagnetism.
He successfully called the dynamic equation with generalized coordinates and generalized momentum as independent variables now Hamiltonian canonical equation. He also established the Hamiltonian function closely related to energy. He explained the phenomenon of conical refraction and contributed to the establishment of modern vector analysis methods. He also founded quaternions. These achievements are widely used in modern physics.
Hamilton's most outstanding achievements in mathematics are differential equations and functional analysis, such as Hamilton operator and Hamilton-Jacobian equation. In addition, his research on wavy surfaces, the supplement to Galois' theory and the introduction of associative laws in mathematics are also his achievements.
Hamilton's two long papers on variational principles and canonical equations are entitled "On a General Method in Dynamics" (1834) and "On a General Method in Dynamics". 1835), all included in the second volume of his mathematics paper (1940).