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Lie Groups, Physics, and Geometry by Robert Gilmore

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16

Lie groups and differential equations

 

 

Lie group theory was initially developed to facilitate the solution of differential equations. In this guise its many powerful tools and results are not extensively known in the physics community. This chapter is designed as an antidote to this anemia. Lie’s methods are an extension of Galois’ methods for algebraic equations to the study of differential equations. The extension is in the spirit of Galois’ work: the technical details are not similar. The principle observation – Lie’s great insight – is that the simple constant that can by added to any indefinite integral of dy/dx = g(x) is in fact an element of a continuous symmetry group – the group that maps solutions of the differential equation ...

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