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x, 185 p. : illustrations ; 25 cm. Content notes : Machine generated contents note: pt. 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 of invariant manifolds --
2.3.3. Proof of Theorem 2.9 --
2.3.4. Examples on Theorem 2.9 --
Exercises to Chapter 2 --
3. Groups admitted 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. Prolongation of group transformations --
3.2.1. One-dimensional case --
3.2.2. Prolongation with several differential variables --
3.2.3. General case --
3.3. Prolongation of group generators --
3.3.1. One-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 --
4.1.1. Commutator --
4.1.2. Properties of the commutator --
4.1.3. Properties of determining equations --
4.1.4. Lie algebras --
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 algebras --
4.3.2. Similar Lie algebras --
4.4. Low-dimensional Lie algebras --
4.4.1. One-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 IR3 --
4.5. Lie algebras and multi-parameter groups --
4.5.1. Definition of multi-parameter groups' --
4.5.2. Construction of multi-parameter groups --
Exercises to Chapter 4 --
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 1 --
pt. II Approximate Transformation Groups --
6. Preliminaries --
6.1. Motivation --
6.2. A sketch on Lie transformation groups --
6.2.1. One-parameter transformation groups --
6.2.2. Canonical parameter --
6.2.3. Group generator and Lie equations --
6.2.4. Exponential map --
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 of one-parameter approximate transformation groups --
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. Examples --
8.3.1. First example --
8.3.2. Approximate commutator and Lie algebras --
8.3.3. Second example --
8.3.4. Third example --
Exercises to Chapter 8 --
9. Applications --
9.1. Integration of equations with a small parameter using approximate symmetries --
9.1.1. Equation haying 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. Lie algebras.

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