Preface
Part Ⅰ Local Transformation Groups
1 Preliminaries
1.1 Changes of frames of reference and point transformations
1.1.1 Translations
1.1.2 Rotations
1.1.3 Galilean transformation
1.2 Introduction of transformation groups
1.2.1 Definitions and examples
1.2.2 Different types of groups
1.3 Some useful groups
1.3.1 Finite continuous groups on the straight line
1.3.2 Groups on the plane
1.3.3 Groups in IRn
Exercises to Chapter 1
2 One-parameter groups and their invariants
2.1 Local groups of transformations
2.1.1 Notation and definition
2.1.2 Groups written in a canonical parameter
2.1.3 Infinitesimal transformations and generators
2.1.4 Lie equations
2.1.5 Exponential map
2.1.6 Determination of a canonical parameter
2.2 Invariants
2.2.1 Definition and infinitesimal test
2.2.2 Canonical variables
2.2.3 Construction of groups using canonical variables
2.2.4 Frequently used groups in the plane
2.3 Invariant equations
2.3.1 Definition and infinitesimal test
2.3.2 Invariant representation ofinvariant manifolds
2.3.3 Proof of Theorem
2.3.4 Examples on Theorem
Exercises to Chapter 2
3 Groups adnutted by differential equations
3.1 Preliminaries
3.1.1 Differential variables and functions
3.1.2 Point transformations
3.1.3 Frame of differential equations
3.2 Ptolongation of group transformations
3.2.1 0ne-dimensional case
3.2.2 Prolongation with several differential variables
3.2.3 General case
3.3 Prolongation of group generators
3.3.1 0ne-dimensional case
3.3.2 Several differential variables
3.3.3 General case
3.4 First definition of symmetry groups
3.4.1 Definition
3.4.2 Examples
3.5 Second definition of symmetry groups
3.5.1 Definition and determining equations
3.5.2 Determining equation for second-order ODEs
3.5.3 Examples on solution of determining equations
Exercises to Chapter 3
4 Lie algebras of operators
4.1 Basic definitions
4.1.2 Properties of the commutator
4.1.3 Properties of determining equations
4.2 Basic properties
4.2.1 Notation
4.2.2 Subalgebra and ideal
4.2.3 Derived algebras
4.2.4 Solvable Lie algebras
4.3 Isomorphism and similarity
4.3.1 Isomorphic Lie akebras
4.3.2 Similar Lie algebras
4.4 Low-dimensionalLie algebras
4.4.1 0ne-dimensional algebras
4.4.2 Two-dimensional algebras in the plane
4.4.3 Three-dimensional algebras in the plane
4.4.4 Three-dimensional algebras in lR3
4.5 Lie algebras and multi-parameter groups
4.5.1 Definition of multi-parameter groups
4.5.2 Construction of multi-parameter groups
5 Galois groups via symmetries
5.1 Preliminaries
5.2 Symmetries of algebraic equations
5.2.1 Determining equation
5.2.2 First example
5.2.3 Second example
5.2.4 Third example
5.3 Construction of Galois groups
5.3.1 First example
5.3.2 Second example
5.3.3 Third example
5.3.4 Concluding remarks
Assignment to Part I
Part II Approximate Transformation Groups
6.1 Motivation
6.2 A sketch on Lie transformation groups
6.2.1 0ne-parameter transformation groups
6.2.2 Canonical parameter
6.2.3 Group generator and Lie equations
6.3 Approximate Cauchy problem
6.3.1 Notation
6.3.2 Definition of the approximate Cauchy problem
7 Approximate transformations
7.1 Approximate transformations defined
7.2 Approximate one-parameter groups
7.2.1 Introductory remark
7.2.2 Definition ofone-parameter approximate
7.2.3 Generator of approximate transformation group
7.3 Infinitesimal description
7.3.1 Approximate Lie equations
7.3.2 Approximate exponential map
Exercises to Chapter 7
8 Approximate symmetries
8.1 Definition of approximate symmetries
8.2 Calculation of approximate symmetries
8.2.1 Determining equations
8.2.2 Stable symmetries
8.2.3 Algorithm for calculation
8.3.2 Approximate commutator and Lie algebras
9.1 Integration of equations with a smallparameter usingapproximate
symmetries
9.1.1 Equation having no exact point symmetries
9.1.2 Utilization of stable symmetries
9.2 Approximately invariant solutions
9.2.1 Nonlinear wave equation
9.2.2 Approximate travelling waves of KdV equation
9.3 Approximate conservation laws
Exercises to Chapter 9
Assignment to Part II
Bibliography
Index
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