Variation of parameters: Difference between revisions

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* 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, [http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f811.image pages 809–816].</ref> 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 asto have been 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.<ref>See:

* Michael Efroimsky (2005) [http://onlinelibrary.wiley.com/doi/10.1196/annals.1370.016/abstract "Gauge Freedom in Orbital Mechanics." ANYAS, Vol. 1065, pp. 346–374 (2005)]

* Michael Efroimsky and Peter Goldreich (2004) [http://www.aanda.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/aa/abs/2004/09/aa0058/aa0058.html "Gauge symmetry of the ''N''-body problem of Celestial Mechanics." Astronomy and Astrophysics, Vol. 415, pp. 1187–1199. (2004)]