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內容簡介

　　《非綫性物理科學：變換群和李代數（英文版）》為作者在俄羅斯、美國、南非和瑞典多年講述變換群和李群分析課程的講義。書中所討論的局部李群方法提供瞭求解非綫性微分方程解析解通用且非常有效的方法，而近似變換群可以提高構造含少量參數的微分方程的技巧。《非綫性物理科學：變換群和李代數（英文版）》通俗易懂、敘述清晰，並提供豐富的模型，能幫助讀者輕鬆地逐步深入各種主題。

作者簡介

　　伊布拉基莫夫（Ibragimov，N.H.），教授，瑞士科學傢，被公認為是在微分方程對稱分析方麵世界上最具權威的專傢之一。他發起並構建瞭現代群分析理論，並推動瞭該理論在多方麵的應用。

內頁插圖

目錄

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
非綫性物理科學：變換群和李代數（英文版） [Nonlinear Physical Science: Transformation Groups and Lie Algebras] 下載 mobi epub pdf txt 電子書

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