Editing Calculus of variations (section)

== History ==

The calculus of variations began with the work of [[Isaac Newton]], such as with [[Newton's minimal resistance problem]], which he formulated and solved in 1685, and later published in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' in 1687,<ref name=":2">{{Cite book |last=Goldstine |first=Herman H. |url=https://books.google.com/books?id=_iTnBwAAQBAJ&pg=PA7 |title=A History of the Calculus of Variations from the 17th Through the 19th Century |date=1980 |publisher=Springer New York |isbn=978-1-4613-8106-8 |series= |location= |pages=7–21}}</ref> which was the first problem in the field to be formulated and correctly solved,<ref name=":2" /> and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.<ref name=":0">{{Citation |last=Ferguson |first=James |title=A Brief Survey of the History of the Calculus of Variations and its Applications |date=2004 |arxiv=math/0402357 |bibcode=2004math......2357F }}</ref><ref name=":1">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA36 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=36–39 |language=en |doi=10.1142/q0108}}</ref><ref>{{Cite journal |last=Torres |first=Delfim F. M. |date=2021-07-29 |title=On a Non-Newtonian Calculus of Variations |journal=Axioms |language=en |volume=10 |issue=3 |pages=171 |doi=10.3390/axioms10030171 |doi-access=free |issn=2075-1680|arxiv=2107.14152 }}</ref> This problem was followed by the [[brachistochrone curve]] problem raised by [[Johann Bernoulli]] (1696),<ref name=GelfandFominP3>{{cite book| last1=Gelfand|first1=I. M.|author-link1=Israel Gelfand|last2=Fomin|first2=S. V.|author-link2=Sergei Fomin|title=Calculus of variations | year=2000|publisher=Dover Publications|location=Mineola, New York|isbn=978-0486414485|page=3| url=https://books.google.com/books?id=YkFLGQeGRw4C|edition=Unabridged repr.|editor1-last=Silverman| editor1-first=Richard A.}}</ref> which was similar to one raised by [[Galileo Galilei]] in 1638, but he did not solve the problem explicity nor did he use the methods based on calculus.<ref name=":0" /> Bernoulli had solved the problem, using the principle of least time in the process, but not calculus of variations, whereas Newton did to solve the problem in 1697, and as a result, he pioneered the field with his work on the two problems.<ref name=":1" /> The problem would immediately occupy the attention of [[Jacob Bernoulli]] and the [[Guillaume de l'Hôpital|Marquis de l'Hôpital]], but [[Leonhard Euler]] first elaborated the subject, beginning in 1733. [[Joseph-Louis Lagrange]] was influenced by Euler's work to contribute greatly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the ''calculus of variations'' in his 1756 lecture ''Elementa Calculi Variationum''.<ref name=Thiele>{{cite book |last=Thiele |first=Rüdiger |editor-last1=Bradley |editor-first1=Robert E. |editor-last2=Sandifer |editor-first2=C. Edward |title=Leonhard Euler: Life, Work and Legacy |publisher=Elsevier |year=2007 |page=249 |chapter=Euler and the Calculus of Variations |chapter-url=https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA249 |isbn=9780080471297}}</ref><ref name=Goldstine>{{cite book |last=Goldstine |first=Herman H. |year=2012 |title=A History of the Calculus of Variations from the 17th through the 19th Century |url=https://books.google.com/books?id=_iTnBwAAQBAJ&q=%22Indeed+after%22&pg=110 |publisher=Springer Science & Business Media |page=110 |isbn=9781461381068 |author-link=Herman Goldstine }}</ref>{{efn|"Euler waited until Lagrange had published on the subject in 1762 ... before he committed his lecture ... to print, so as not to rob Lagrange of his glory. Indeed, it was only Lagrange's method that Euler called Calculus of Variations."<ref name=Thiele/>}}

[[Adrien-Marie Legendre]] (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. [[Isaac Newton]] and [[Gottfried Leibniz]] also gave some early attention to the subject.<ref name="brunt">{{cite book |last=van Brunt |first=Bruce |title=The Calculus of Variations |publisher=Springer |year=2004 |isbn=978-0-387-40247-5}}</ref> To this discrimination [[Vincenzo Brunacci]] (1810), [[Carl Friedrich Gauss]] (1829), [[Siméon Denis Poisson|Siméon Poisson]] (1831), [[Mikhail Ostrogradsky]] (1834), and [[Carl Gustav Jacob Jacobi|Carl Jacobi]] (1837) have been among the contributors. An important general work is that of [[Pierre Frédéric Sarrus]] (1842) which was condensed and improved by [[Augustin-Louis Cauchy]] (1844). Other valuable treatises and memoirs have been written by [[Strauch]]{{which|date=October 2024}} (1849), [[John Hewitt Jellett]] (1850), [[Otto Hesse]] (1857), [[Alfred Clebsch]] (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of [[Karl Weierstrass]]. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The [[Hilbert's twentieth problem|20th]] and the [[Hilbert's twenty-third problem|23rd]] [[Hilbert problems|Hilbert problem]] published in 1900 encouraged further development.<ref name="brunt" />

In the 20th century [[David Hilbert]], [[Oskar Bolza]], [[Gilbert Ames Bliss]], [[Emmy Noether]], [[Leonida Tonelli]], [[Henri Lebesgue]] and [[Jacques Hadamard]] among others made significant contributions.<ref name="brunt" /> [[Marston Morse]] applied calculus of variations in what is now called [[Morse theory]].<ref name="ferguson">{{cite arXiv |last=Ferguson |first=James |eprint=math/0402357 |title= Brief Survey of the History of the Calculus of Variations and its Applications |year=2004 }}</ref> [[Lev Pontryagin]], [[R. Tyrrell Rockafellar|Ralph Rockafellar]] and F. H. Clarke developed new mathematical tools for the calculus of variations in [[optimal control theory]].<ref name="ferguson" /> The [[dynamic programming]] of [[Richard Bellman]] is an alternative to the calculus of variations.<ref>[[Dimitri Bertsekas]]. Dynamic programming and optimal control. Athena Scientific, 2005.</ref><ref name="bellman">{{cite journal |last=Bellman |first=Richard E. |title= Dynamic Programming and a new formalism in the calculus of variations |year=1954 |journal= Proc. Natl. Acad. Sci. | issue=4 | pages=231–235|pmc=527981 |pmid=16589462 |volume=40 |doi=10.1073/pnas.40.4.231|bibcode=1954PNAS...40..231B |doi-access=free }}</ref><ref>{{cite web |title=Richard E. Bellman Control Heritage Award |year=2004 |url=http://a2c2.org/awards/richard-e-bellman-control-heritage-award |work=American Automatic Control Council |access-date=2013-07-28 |archive-date=2018-10-01 |archive-url=https://web.archive.org/web/20181001032837/http://a2c2.org/awards/richard-e-bellman-control-heritage-award |url-status=dead }}</ref>{{efn|See '''[[Harold J. Kushner]] (2004)''': regarding Dynamic Programming, "The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community."}}