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**Introduction to symmetry analysis.**
*(English)*
Zbl 1082.34001

Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (ISBN 0-521-77740-2/pbk; 0-521-77183-8/hbk). xli, 612 p. (2002).

Summary: This textbook is designed mainly for first- and second-year graduate students in science, engineering and applied mathematics, although the material is presented in a form that should be understandable to an upper-level undergraduate with a background in differential equations. The main goal is to teach methods of symmetry analysis and to instill in the student a sense of confidence in dealing with complex problems. The central theme is that anytime one is confronted with a physical problem and a set of equations to solve, the first step is to study the problem using dimensional analysis and the second one is to use the methods of symmetry analysis to work out the Lie groups (symmetries) of the governing equations. This may or may not produce a simplification, but it will almost always bring clarity to the problem. It is the author’s firm belief that symmetry analysis should be as familiar to the student as Fourier analysis.

Most of the theory, with a large number of relatively short worked examples, is developed in the first part of the book, while the second part contains a number of fully worked problems where the role of symmetry analysis as a part of the complete solution is illustrated. The emphasis is on applications, and the exercises provided at the end of each chapter are designed to help the reader practice the material.

In more detail, the book is organized as follows. The exposition starts with a historical preface. Chapters 1–3 and 5–10 cover a standard course: introduction to symmetry, dimensional analysis, introduction to systems of ODEs and first-order PDEs, state-space analysis, introduction to one-parameter Lie groups, symmetry analysis of first-order ODEs, introduction to differential functions, symmetry analysis of higher-order ODEs and PDEs, introduction to laminar boundary layers. Chapters 4 and 11–16 contain more advanced material: classical dynamics, incompressible flow, compressible flow, similarity rules for turbulent shear flows, Lie-Bäcklund transformations, variational symmetries and conservation laws, Bäcklund transformations and nonlocal groups. To make the book more complete some background material of advanced nature is presented in Appendices 1–3.

The first exercises involving the identification of Lie symmetries should be worked by hand so that the reader has a chance to practice the Lie algorithm, but it becomes quickly apparent that the calculations are huge even in fairly simple cases. This is one of the main reasons why Lie theory was never adopted in the mainstream curricula in science and engineering. Fortunately, we now live in an era when powerful symbol manipulation software packages are widely available. A special MATHEMATICA-based package was developed by the author of the book and it is included on CD along with more than sixty sample notebooks that are carefully coordinated with the examples and exercises in the text. Details of the package are described in Appendix 4.

Most of the theory, with a large number of relatively short worked examples, is developed in the first part of the book, while the second part contains a number of fully worked problems where the role of symmetry analysis as a part of the complete solution is illustrated. The emphasis is on applications, and the exercises provided at the end of each chapter are designed to help the reader practice the material.

In more detail, the book is organized as follows. The exposition starts with a historical preface. Chapters 1–3 and 5–10 cover a standard course: introduction to symmetry, dimensional analysis, introduction to systems of ODEs and first-order PDEs, state-space analysis, introduction to one-parameter Lie groups, symmetry analysis of first-order ODEs, introduction to differential functions, symmetry analysis of higher-order ODEs and PDEs, introduction to laminar boundary layers. Chapters 4 and 11–16 contain more advanced material: classical dynamics, incompressible flow, compressible flow, similarity rules for turbulent shear flows, Lie-Bäcklund transformations, variational symmetries and conservation laws, Bäcklund transformations and nonlocal groups. To make the book more complete some background material of advanced nature is presented in Appendices 1–3.

The first exercises involving the identification of Lie symmetries should be worked by hand so that the reader has a chance to practice the Lie algorithm, but it becomes quickly apparent that the calculations are huge even in fairly simple cases. This is one of the main reasons why Lie theory was never adopted in the mainstream curricula in science and engineering. Fortunately, we now live in an era when powerful symbol manipulation software packages are widely available. A special MATHEMATICA-based package was developed by the author of the book and it is included on CD along with more than sixty sample notebooks that are carefully coordinated with the examples and exercises in the text. Details of the package are described in Appendix 4.

Reviewer: Victor V. Zharinov (MR 1929892)

### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

00A05 | Mathematics in general |

34C14 | Symmetries, invariants of ordinary differential equations |

34A26 | Geometric methods in ordinary differential equations |

34-04 | Software, source code, etc. for problems pertaining to ordinary differential equations |

35-04 | Software, source code, etc. for problems pertaining to partial differential equations |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

76M60 | Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics |