Lectures on the theory of group properties of differential equations. Edited by Nail H. Ibragimov (Q2842214)

This book is a compilation of lectures on the group properties of ordinary differential equations given by the author (L.V. Ovsyannikov) for undergraduate students in the course year 1965/66. The aim of the book is threefold: to present in a clear and systematic way the Lie theory, to connect it with ``modern results'' and to present several physical examples as the heat equation and gasdynamic equations. This translation and compilation of the lectures, done by Nail H. Ibragimov, offers a self contained and well organized material for self study and reference.NEWLINENEWLINEThe book is organized in three chapters. The first chapter deals with one-parameter group transformations, first for an Euclidean space of dimension \(N\) and then for differential equations. It contains all the definitions and results of classical Lie theory as the definition of invariants, the first and second prolongation of the group operator and the determining equations of the groups admitted by a differential equation. Hence, special attention is paid to first- and second-order ordinary differential equations. The text is full of examples, including the aforementioned physical equations which are considered along the book.NEWLINENEWLINEThe second chapter deals with Lie algebras and Lie groups and it contains the three fundamental theorems of Lie and the description of the local Lie groups of transformations. The Lie parenthesis and the notion of commutators are introduced. The third chapter is about solutions of differential equations which are invariant for the associated local group and how these solutions allow to reduce the differential equation. To end with, several problems are stated and some of them have been solved during the last decades.NEWLINENEWLINENo result of the book can be highlighted among the others as all the results are important, correlated and they belong to a classical theory. Lie theory has got a lot of repercussion in physics and this is one of the reasons why a researcher can be interested in the book. Indeed, this theory gathers many results of ordinary differential equations with a systematic and rigorous point of view. Normal form theory, integrability, isochronicity, invariant manifolds, and a long et cetera are examples of this fact.