Sophus Lie, the mathematician
Sophus Lie, the mathematician
Sophus Lie, the mathematician
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(F,g) = 0which enters in Jacobi's work on solving (1) becomes for <strong>Lie</strong> <strong>the</strong> equation[X 9 ,XF] = 0,which leads to a search for an infinitesimal contact ransformation leavinginvariant <strong>the</strong> giv'en equation (1). This signals a shift in his worktowards transformation groups and differential equations.After <strong>Lie</strong> had applied for a professorship in Lund <strong>the</strong> Norwegianstorting approved a Professorship for him in Christiania, July 1, 1872.He continued his work on contact transformations but <strong>the</strong>n in 1873started on a systematic study of transformation groups. The motivationis <strong>the</strong> question, stated explicitly in a paper [20] from 1874: "Howcan <strong>the</strong> knowledge of a stability group for a differential equation be utilizedtowards its integration?" A point transformation is said to leavea differential equation stable if it permutes <strong>the</strong> solutions. Here <strong>Lie</strong> is ofcourse inspired by Galois <strong>the</strong>ory for algebraic equations which he hadheard about already in Sylow's lecture in 1862, and had presumablydiscussed with Jordan in Paris, who had just <strong>the</strong>n (1869) publishedhis clarification of Galois' <strong>the</strong>ory. In <strong>the</strong> quoted paper <strong>Lie</strong> proves <strong>the</strong>following now famous <strong>the</strong>orem, stated below.Considerdy Y(x, y)dx X(x, y)and a local one-parameter subgroupinduced vector fieldt of diffeomorphisms of R 2 witht= t=(p)/ + i ; -J/ay.Theorem. The transformations t leave <strong>the</strong> equation stable if andonly if <strong>the</strong> vector field Z = Xa/&z + Yalay satisfies[I, Z] = AZ(A a function).In this case (X7 7 -Y)- is an integrating factor to <strong>the</strong> equationX dy - Y dx = 0.11