Editing William Rowan Hamilton (section)

===Context and importance of the work===
[[Hamiltonian mechanics]] was a powerful new technique for working with [[equations of motion]]. Hamilton's advances enlarged the class of mechanical problems that could be solved. His principle of "Varying Action" was based on the [[calculus of variations]], in the general class of problems included under the [[principle of least action]] which had been studied earlier by [[Maupertuis|Pierre Louis Maupertuis]], [[Euler]], [[Joseph Louis Lagrange]] and others. Hamilton's analysis uncovered a deeper mathematical structure than had been previously understood, in particular a symmetry between momentum and position. The credit for discovering what are now called the [[Lagrangian (field theory)|Lagrangian]] and [[Lagrange's equations]] also belongs to Hamilton.

Both the [[Lagrangian mechanics]] and Hamiltonian approaches have proven important in the study of continuous classical systems in physics, and quantum mechanical systems: the techniques find use in [[electromagnetism]], [[quantum mechanics]], [[relativity theory]] and [[quantum field theory]]. In the ''[[Dictionary of Irish Biography]]'' [[David Spearman]] writes:<ref>[https://dib.cambridge.org/home.do Dictionary of Irish Biography: Hamilton, William Rowan] {{Webarchive|url=https://web.archive.org/web/20190406170328/https://dib.cambridge.org/home.do |date=6 April 2019 }} [[Cambridge University Press]]</ref>

{{blockquote|The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated. The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this the most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting point for calculation in almost any area of physics.}}

Many scientists, including [[Joseph Liouville|Liouville]], [[C. G. J. Jacobi|Jacobi]], [[Jean Gaston Darboux|Darboux]], [[Henri Poincaré|Poincaré]], [[Kolmogorov]], [[Ilya Prigogine|Prigogine]]<ref>{{Cite journal|last1=Petrosky|first1=T|last2=Prigogine|first2=Ilya|title=The extension of classical dynamics for unstable Hamiltonian systems|journal=Computers & Mathematics with Applications|year=1997|volume=34|issue=2–4|pages=1–44|doi=10.1016/S0898-1221(97)00116-8|language=en|doi-access=free}}</ref> and [[V. I. Arnold|Arnold]], have extended Hamilton's work, in [[mechanics]], [[differential equations]] and [[symplectic geometry]].<ref>{{Cite web|last=Hartnett|first=Kevin|title=How Physics Found a Geometric Structure for Math to Play With|url=https://www.quantamagazine.org/how-physics-gifted-math-with-a-new-geometry-20200729/|access-date=2020-07-30|website=Quanta Magazine|date=29 July 2020|language=en|archive-date=29 July 2020|archive-url=https://web.archive.org/web/20200729163345/https://www.quantamagazine.org/how-physics-gifted-math-with-a-new-geometry-20200729/|url-status=live}}</ref>