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In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and don't work for all inhomogeneous linear differential equations. Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa. == History == The method of variation of parameters was introduced by the Swiss-born mathematician Leonhard Euler (1707–1783) and completed by the Italian-French mathematician Joseph-Louis Lagrange (1736–1813).〔See: * Forest Ray Moulton, ''An Introduction to Celestial Mechanics'', 2nd ed. (first published by the Macmillan Company in 1914; reprinted in 1970 by Dover Publications, Inc., Mineola, New York), (page 431 ). * Edgar Odell Lovett (1899) ("The theory of perturbations and Lie's theory of contact transformations," ) ''The Quarterly Journal of Pure and Applied Mathematics'', vol. 30, pages 47-149; see especially pages 48-61.〕 A forerunner of the method of variation of a celestial body's orbital elements appeared in Euler's work in 1748, while he was studying the mutual perturbations of Jupiter and Saturn.〔Euler, L. (1748) ("Recherches sur la question des inégalités du mouvement de Saturne et de Jupiter, sujet proposé pour le prix de l'année 1748, par l’Académie Royale des Sciences de Paris" ) (on the question of the differences in the movement of Saturn and Jupiter; this subject proposed for the prize of 1748 by the Royal Academy of Sciences (Paris) ) (Paris, France: G. Martin, J.B. Coignard, & H.L. Guerin, 1749).〕 In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements;〔Euler, L. (1749) ("Recherches sur la précession des équinoxes, et sur la nutation de l’axe de la terre," ) ''Histoire'' (''Mémoires'' ) ''de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 289-325 (in 1751 ).〕 and in 1753 he applied the method to his study of the motions of the moon.〔Euler, L. (1753) (Theoria motus lunae: exhibens omnes ejus inaequalitates ... ) (theory of the motion of the moon: demonstrating all of its inequalities ... ) (Saint Petersburg, Russia: Academia Imperialis Scientiarum Petropolitanae (Academy of Science (St. Petersburg) ), 1753).〕 Lagrange first used the method in 1766.〔Lagrange, J.-L. (1766) (“Solution de différens problèmes du calcul integral,” ) ''Mélanges de philosophie et de mathématique de la Société royale de Turin'', vol. 3, pages 179-380.〕 Between 1778 and 1783, Lagrange further developed the method both in a series of memoirs on variations in the motions of the planets〔See: * Lagrange, J.-L. (1781) ("Théorie des variations séculaires des élémens des Planetes. Premiere partie, ... ," ) ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 199-276. * Lagrange, J.-L. (1782) ("Théorie des variations séculaires des élémens des Planetes. Seconde partie, ... ," ) ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 169-292. * Lagrange, J.-L. (1783) ("Théorie des variations périodiques des mouvemens des Planetes. Premiere partie, ... ," ) ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 161-190.〕 and in another series of memoirs on determining the orbit of a comet from three observations.〔See: * Lagrange, J.-L. (1778) ("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 (in 1780 ). * Lagrange, J.-L. (1778) ("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 (in 1780 ). * Lagrange, J.-L. (1783) ("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 (in 1785 ).〕 (It should be noted that 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.〔Michael Efroimsky (2002) ("Implicit gauge symmetry emerging in the N-body problem of celestial mechanics," ) page 3.〕) During 1808-1810, Lagrange gave the method of variation of parameters its final form in a series of papers.〔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, (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, ( pages 771-805 ). * Lagrange, J.-L. (1810) “Second mémoire 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, (pages 809-816 ).〕 The central result of his study was the system of planetary equations in the form of Lagrange, which described the evolution of the Keplerian parameters (orbital elements) of a perturbed orbit. In his description of evolving orbits, Lagrange set a reduced two-body problem as an unperturbed solution, and presumed that all perturbations come from the gravitational pull which the bodies other than the primary exert at the secondary (orbiting) body. Accordingly, his method implied that the perturbations depend solely on the position of the secondary, but not on its velocity. In the 20th century, celestial mechanics began to consider interactions which depend on both positions and velocities (relativistic corrections, atmospheric drag, inertial forces). Therefore, the method of variation of parameters used by Lagrange was extended to the situation with velocity-dependent forces.〔See: * Michael Efroimsky (2005) ("Gauge Freedom in Orbital Mechanics." ANYAS, Vol. 1065, pp. 346–374 (2005) ) * Michael Efroimsky and Peter Goldreich (2004) ("Gauge symmetry of the N-body problem of Celestial Mechanics." Astronomy and Astrophysics, Vol. 415, pp. 1187–1199. (2004) ) * Michael Efroimsky and Peter Goldreich (2003) ("Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach." Journal of Mathematical Physics, Vol. 44, pp. 5958–5977. (2003) )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Variation of parameters」の詳細全文を読む スポンサード リンク
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