Jacobi and the Analytical Origins of Lie’s Theory

Abstract

Lie began to study the general theory of first-order partial differential equations assiduously while in Paris in 1870. His main source was a well-written monographic essay on the subject by V. G, Imschenetsky [1869]. Imschenet-sky’s essay, which ran to almost two hundred pages, was symptomatic of the interest in the theory of first order equations that had been generated by the contributions of Jacobi. Although Jacobi had developed his ideas in the 1830s (he died in 1851) and lectured on their relation to dynamical systems in the early 1840s, it was not until the 1860s that most of this work was published by Clebsch [Jacobi 1862; 1866]. The appearance of these papers stimulated considerable interest in the theory of first order partial differential equations, particularly in France and Germany. By the time he was in Paris, Lie was already interested in the application of groups to differential equations, for Klein’s observations (discussed in Section 1.3) of an analogy between some of his work on tetrahedral complexes and Galois theory had been made in Berlin in 1869.