Revision as of 14:39, 18 March 2020 edit Strausshilbert (talk \| contribs) 10 edits We are seeking the functions A and B such that Au1+Bu2 is a "particular", not "general", solution to the non-homogeneous equation. General solution=solution to homogeneous equation+particular solution to non-homogeneous equation.m ← Previous edit		Revision as of 20:57, 31 May 2020 edit undo Bender235 (talk \| contribs) Autopatrolled, Extended confirmed users, Pending changes reviewers, Rollbackers, Template editors 472,824 edits m →History: Replaced arXiv PDF link with more mobile-friendly abstract link, replaced: https://arxiv.org/pdf/ → https://arxiv.org/abs/ Tag: AWB Next edit →
Line 44: * Lagrange, J.-L. (1778) [https://books.google.com/books?id=F90_AAAAYAAJ&pg=PA60-IA55#v=onepage&q&f=false "Sur le probleme de la détermination des orbites des cometes d'après trois observations, premier mémoire"] (On the problem of determining the orbits of comets from three observations, first memoir), ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 111–123 [published in 1780]. * Lagrange, J.-L. (1778) [https://books.google.com/books?id=F90_AAAAYAAJ&pg=PA60-IA68#v=onepage&q&f=false "Sur le probleme de la détermination des orbites des cometes d'après trois observations, second mémoire"], ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 124–161 [published in 1780]. * Lagrange, J.-L. (1783) [http://gallica.bnf.fr/ark:/12148/bpt6k229223s/f498.image "Sur le probleme de la détermination des orbites des cometes d'après trois observations. Troisième mémoire, dans lequel on donne une solution directe et générale du problème."], ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 296–332 [published in 1785].</ref> (Euler and Lagrange applied this method to nonlinear differential equations and that, instead of varying the coefficients of linear combinations of solutions to homogeneous equations, they varied the constants of the unperturbed motions of the celestial bodies.<ref>Michael Efroimsky (2002) [https://arxiv.org/~~pdf~~abs/astro-ph/0212245~~.pdf~~ "Implicit gauge symmetry emerging in the ''N''-body problem of celestial mechanics,"] page 3.</ref>) During 1808–1810, Lagrange gave the method of variation of parameters its final form in a series of papers.<ref>See: * Lagrange, J.-L. (1808) “Sur la théorie des variations des éléments des planètes et en particulier des variations des grands axes de leurs orbites,” ''Mémoires de la première Classe de l’Institut de France''. Reprinted in: Joseph-Louis Lagrange with Joseph-Alfred Serret, ed., ''Oeuvres de Lagrange'' (Paris, France: Gauthier-Villars, 1873), vol. 6, [http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f715.image pages 713–768]. * Lagrange, J.-L. (1809) “Sur la théorie générale de la variation des constantes arbitraires dans tous les problèmes de la méchanique,” ''Mémoires de la première Classe de l’Institut de France''. Reprinted in: Joseph-Louis Lagrange with Joseph-Alfred Serret, ed., ''Oeuvres de Lagrange'' (Paris, France: Gauthier-Villars, 1873), vol. 6, [http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f773 pages 771–805].

Variation of parameters: Difference between revisions