Overview

Analytical development of the variation of parameters was first given by Leonhard Euler in a series of memoirs on the mutual perturbations of Jupiter and Saturn for which he received the prizes of the French Academy in the years 1748 and 1752. The method is also called the variation of orbital elements or the variation of constants the latter referring to integration constants. Euler's treatment of the method of variation of parameters was not entirely general since he did not consider the orbital elements as being simultaneously variable. It is noteworthy, however, that the first steps in the expansion of the disturbing function were made by Euler in those papers.

Joseph-Louis Lagrange wrote his first memoir on the perturbations of Jupiter and Saturn in 1766 in which he made further advances in the variation of parameters method. His final equations were still incorrect because he regarded the major axes and the times of perihelion passage as constants. However, his expressions for the angle of inclination, the longitude of the ascending node, and the argument of perihelion were all perfectly correct. Later, in 1782, he developed completely and for the first time the method of the variation of parameters in a prize memoir on the perturbations of comets moving in elliptical orbits. One of the objectives in this chapter is the derivation of Lagrange's planetary equations.

The most dramatic application of the method was made independently and almost simultaneously by the Englishman John Couch Adams (1819 1892) and the Frenchman Urbain-Jean-Joseph...