Algorithmic Lie Theory for Solving Ordinary Differential Equations
When Sophus Lie developed his theory of groups of continuous transformations (nowadays called Lie groups), his aim was to find a systematic way to integrate differential equations. The theory of Lie groups and Lie algebras was systematically developed into a broad field of mathematics, while the progress in Lie's original goal was much slower. A part of the problem was connected with a need for very complicated analytic computations, which can now be done more efficiently using computer algebra software. The aim of this book is to discuss algorithms for solving ordinary differential equations using the Lie approach. The author first introduces basic tools (the Janet algorithm for solving linear partial differential equations, transformation groups, equivalence and invariants of differential equations). The main part of the book is devoted to ordinary differential equations of at most third order, to a study of symmetries of a given equation, its transformation to canonical form and finally to algorithms for solutions. There is a lot of exercises, some of them with solutions. Software for computer calculations is available on a webpage.