այն պարզեցնելով Լայբնիցը հաջորդ տարի գտավ լուծումները։f<ref>{{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}</ref>
HistoricallyՊատմականորեն, theերաժշտական problemգործիքի ofթրթռացող aլարի vibratingպրոբլեմը stringուսումնասիրվել such as that of aէ [[musicalԺան instrument]]Լը wasՌոն studiedԴ'Ալամբեր|Ժան byԼը [[JeanՌոն le Rond d'AlembertԴ’Ալամբերի]], [[LeonhardԼեոնարդ EulerԷյլեր|Լեոնարդ Էյլերի]], [[DanielԴանիել BernoulliԲեռնուլի|Դանիել Բեռնուլիի]],andև [[Joseph-LouisԺոզեֆ LagrangeԼուի Լագրանժ|Ժոզեֆ Լուի Լագրանժի]]. կողմից։ <ref>{{cite journal|url = http://homes.chass.utoronto.ca/~cfraser/vibration.pdf |title = Review of ''The evolution of dynamics, vibration theory from 1687 to 1742'', by John T. Cannon and Sigalia Dostrovsky|last= Frasier|first=Craig|journal=Bulletin (New Series) of the American Mathematical Society |date=July 1983 |volume= 9| issue = 1}}</ref><ref>{{cite journal |first=Gerard F. |last=Wheeler |first2=William P. |last2=Crummett |title=The Vibrating String Controversy |journal= [[American Journal of Physics|Am. J. Phys.]] |year=1987 |volume=55 |issue=1 |pages=33–37 |doi=10.1119/1.15311 |bibcode = 1987AmJPh..55...33W }}</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18 Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref> In 1746, d’Alembert discovered the one-dimensional [[wave equation]], and within ten years Euler discovered the three-dimensional wave equation.<ref name=Speiser>Speiser, David. ''[https://books.google.com/books?id=9uf97reZZCUC&pg=PA191 Discovering the Principles of Mechanics 1600-1800]'', p. 191 (Basel: Birkhäuser, 2008).</ref>
The [[Euler–Lagrange equation]] was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to [[mechanics]], which led to the formulation of [[Lagrangian mechanics]].
