Who came up with the Euler-Lagrange equation? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:24:51Z http://mathoverflow.net/feeds/question/103623 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103623/who-came-up-with-the-euler-lagrange-equation Who came up with the Euler-Lagrange equation? jacobi 2012-07-31T16:58:35Z 2012-07-31T20:18:51Z <p>Which man came up with the solution to the basic Calculus of Variations problem first?</p> <p><a href="http://en.wikipedia.org/wiki/Euler-Lagrange" rel="nofollow">http://en.wikipedia.org/wiki/Euler-Lagrange</a></p> <p>Makes it sound like Lagrange got it first in 1755, then sent it to Euler.</p> <p>But</p> <p><a href="http://en.wikipedia.org/wiki/Calculus_of_variations" rel="nofollow">http://en.wikipedia.org/wiki/Calculus_of_variations</a></p> <p>makes it sound like Euler got it in the 1730s.</p> <p>Could somebody please explain which man actually wrote out the equation first, and what the other then contributed to it? </p> http://mathoverflow.net/questions/103623/who-came-up-with-the-euler-lagrange-equation/103626#103626 Answer by Jon for Who came up with the Euler-Lagrange equation? Jon 2012-07-31T18:26:11Z 2012-07-31T18:26:11Z <p>I think the main reference here is <a href="http://eulerarchive.maa.org/date-present.html" rel="nofollow">Euler's archive</a>. You have to look at the two books on mechanics firstly presented on 1736. Euler often presented his works to the community before publication. Being these authors almost contemporary, it is possible that the name to these equations is indeed the right one.</p> http://mathoverflow.net/questions/103623/who-came-up-with-the-euler-lagrange-equation/103629#103629 Answer by Robert Bryant for Who came up with the Euler-Lagrange equation? Robert Bryant 2012-07-31T19:59:35Z 2012-07-31T20:05:38Z <p>According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his <em>Methodus inveniendi</em> [2], Chapter 2, no. 21. Quite often, one speaks of Lagrange's differential equation, or the Euler-Lagrange equations. Yet Lagrange himself attributes this equation to Euler: '<em>Cette équation est quelle qu'Euler a trouvée le premier</em>' (Oeuvres [12], Vol. 10, p. 397)."</p> <p>NB: Euler's <em>Methodus inveniendi</em> was published in 1744. </p> <p>Further down the same page in G&amp;H, one finds "With Lagrange's letter of August 12, 1755 to Euler, the development of the variational calculus took a new turn. Lagrange explained the $\delta$-symbolism which quickly leads to Euler's equation. Euler was very impressed by the new method, and already one year later he lectured about it at the Berlin Academy."</p> http://mathoverflow.net/questions/103623/who-came-up-with-the-euler-lagrange-equation/103633#103633 Answer by Igor Rivin for Who came up with the Euler-Lagrange equation? Igor Rivin 2012-07-31T20:18:51Z 2012-07-31T20:18:51Z <p>Since <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html" rel="nofollow">the brachistochrone problem</a> was solved (by Newton first, but then some others), I would say that the germ of the idea goes to Newton's paper in 1697. In the linked article, Lagrange attributes the general formulation to Euler.</p>