Jacobi and the birth of Lie's theory of groups

The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808–1810

“…Discovered by the German mathematician Carl Gustav Jacob Jacobi (1804-1851) [8,10], the identity that bears his name involves three arbirary smooth functions f , g and h defined on a symplectic (or a Poisson) manifold,…”

Application of Lie group analysis to a core group model for sexually transmitted diseases

“…the idée fixe guiding Lie's work was the development of a Galois theory of differential equations". Also Hawkins had established "the nature and extent of Jacobi's influence upon Lie" [9]. This is particularly noteworthy since 2004 marked two hundred years since Jacobi's birth.…”

“…"Given the fact that the Jacobi Identity is fundamental to the theory of Lie groups, Jacobi's influence upon Lie will come as no surprise. But the bald fact that he inherited the Identity from Jacobi fails to convey fully or accurately the historical dimension of the impact of Jacobi's work on partial differential equations" [9]. Lie's monumental work on transformation groups, [18], [19] and [20], and in particular contact transformations [21], has provided systematic techniques for obtaining exact solutions of differential equations [22].…”

On the inverse problem of calculus of variations for fourth-order equations

“…The Norwegian Sophus Lie, who carefully studied Jacobi's work (Hawkins 1991), found a connection between his groups of transformations and the Jacobi last multiplier (Lie 1874(Lie , 1912. The Italian Bianchi presented Lie's work in his lectures on finite continuous groups of transformations, and described quite clearly the Jacobi last multiplier and its properties (Bianchi 1918).…”