Paperback
Published: 8th October 1990
ISBN: 9780521366892
Number Of Pages: 276
Earn 126 Qantas Points
on this Book
In many branches of physics, mathematics, and engineering, solving a problem means solving a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The emphasis in this text is on how to find and use the symmetries; this is supported by many examples and more than 100 exercises. This book will form an introduction accessible to beginning graduate students in physics, applied mathematics, and engineering. Advanced graduate students and researchers in these disciplines will find the book a valuable reference.
"...as an account of classical general relativity, this well produced and excellently translated book has many virtues." Physics Bulletin "A nice introduction to the theory and practice of finding and using symmetries to solve differential equations." American Mathematical Monthly "...Stephani's book does a good job of motivating the study of Lie group methods for differential equations from an elementary standpoint." SIAM Reviews "The author, who has an easy-to-read practical style, consistently keeps the emphasis on applications. Thus, most physicists will be able to get useful information about ordinary differential equations (ODE's) and partial differential equations (PDE's), without being bogged down in cumbersome mathematical formalism...well worth reading if one is at all interested in sophisticated and powerful symmetry techniques for handling differential equations and if one wishes to have the most straightforward approach to the topic." D. E. Vincent, Physics in Canada "...Stephani...has built a book that tries to guide its readers toward a sure knowledge of this very important tool for finding solutions of (nonlinear) differential equations. In the early sections, the derivations presented are the most clear and detailed ones that this writer has ever seen...Students new to this area will find reading or studying the current book an altogether enjoyable occupation, without any of the intimidation that sometimes is caused by similar books." J.D. Finley, Foundations of Physics
Preface | p. xi |
Introduction | p. 1 |
Ordinary differential equations | |
Point transformations and their generators | p. 5 |
One-parameter groups of point transformations and their infinitesimal generators | p. 5 |
Transformation laws and normal forms of generators | p. 9 |
Extensions of transformations and their generators | p. 11 |
Multiple-parameter groups of transformations and their generators | p. 14 |
Exercises | p. 16 |
Lie point symmetries of ordinary differential equations: the basic definitions and properties | p. 17 |
The definition of a symmetry: first formulation | p. 17 |
Ordinary differential equations and linear partial differential equations of first order | p. 20 |
The definition of a symmetry: second formulation | p. 22 |
Summary | p. 25 |
Exercises | p. 25 |
How to find the Lie point symmetries of an ordinary differential equation | p. 26 |
Remarks on the general procedure | p. 26 |
The atypical case: first order differential equations | p. 27 |
Second order differential equations | p. 28 |
Higher order differential equations. The general nth order linear equation | p. 33 |
Exercises | p. 36 |
How to use Lie point symmetries: differential equations with one symmetry | p. 37 |
First order differential equations | p. 37 |
Higher order differential equations | p. 39 |
Exercises | p. 45 |
Some basic properties of Lie algebras | p. 46 |
The generators of multiple-parameter groups and their Lie algebras | p. 46 |
Examples of Lie algebras | p. 49 |
Subgroups and subalgebras | p. 51 |
Realizations of Lie algebras. Invariants and differential invariants | p. 53 |
Nth order differential equations with multiple-parameter symmetry groups: an outlook | p. 57 |
Exercises | p. 58 |
How to use Lie point symmetries: second order differential equations admitting a G[subscript 2] | p. 59 |
A classification of the possible subcases, and ways one might proceed | p. 59 |
The first integration strategy: normal forms of generators in the space of variables | p. 62 |
The second integration strategy: normal forms of generators in the space of first integrals | p. 66 |
Summary: Recipe for the integration of second order differential equations admitting a group G[subscript 2] | p. 69 |
Examples | p. 70 |
Exercises | p. 74 |
Second order differential equations admitting more than two Lie point symmetries | p. 75 |
The problem: groups that do not contain a G[subscript 2] | p. 75 |
How to solve differential equations that admit a G[subscript 3] IX | p. 76 |
Example | p. 78 |
Exercises | p. 79 |
Higher order differential equations admitting more than one Lie point symmetry | p. 80 |
The problem: some general remarks | p. 80 |
First integration strategy: normal forms of generators in the space(s) of variables | p. 81 |
Second integration strategy: normal forms of generators in the space of first integrals. Lie's theorem | p. 83 |
Third integration strategy: differential invariants | p. 87 |
Examples | p. 88 |
Exercises | p. 92 |
Systems of second order differential equations | p. 93 |
The corresponding linear partial differential equation of first order and the symmetry conditions | p. 93 |
Example: the Kepler problem | p. 96 |
Systems possessing a Lagrangian: symmetries and conservation laws | p. 97 |
Exercises | p. 100 |
Symmetries more general than Lie point symmetries | p. 101 |
Why generalize point transformations and symmetries? | p. 101 |
How to generalize point transformations and symmetries | p. 103 |
Contact transformations | p. 105 |
How to find and use contact symmetries of an ordinary differential equation | p. 107 |
Exercises | p. 109 |
Dynamical symmetries: the basic definitions and properties | p. 110 |
What is a dynamical symmetry? | p. 110 |
Examples of dynamical symmetries | p. 112 |
The structure of the set of dynamical symmetries | p. 114 |
Exercises | p. 116 |
How to find and use dynamical symmetries for systems possessing a Lagrangian | p. 117 |
Dynamical symmetries and conservation laws | p. 117 |
Example: L = (x[superscript 2] + y[superscript 2])/2 - a(2y[superscript 3] + x[superscript 2]y), a [characters not producible] 0 | p. 119 |
Example: the Kepler problem | p. 121 |
Example: geodesics of a Riemannian space - Killing vectors and Killing tensors | p. 123 |
Exercises | p. 127 |
Systems of first order differential equations with a fundamental system of solutions | p. 128 |
The problem | p. 128 |
The answer | p. 129 |
Examples | p. 131 |
Systems with a fundamental system of solutions and linear systems | p. 134 |
Exercises | p. 137 |
Partial differential equations | |
Lie point transformations and symmetries | p. 141 |
Introduction | p. 141 |
Point transformations and their generators | p. 142 |
The definition of a symmetry | p. 145 |
Exercises | p. 147 |
How to determine the point symmetries of partial differential equations | p. 148 |
First order differential equations | p. 148 |
Second order differential equations | p. 154 |
Exercises | p. 160 |
How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry transformations | p. 161 |
The structure of the set of symmetry generators | p. 161 |
What can symmetry transformations be expected to achieve? | p. 163 |
Generating solutions by finite symmetry transformations | p. 164 |
Generating solutions (of linear differential equations) by applying the generators | p. 167 |
Exercises | p. 169 |
How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variables | p. 170 |
The problem | p. 170 |
Similarity variables and how to find them | p. 171 |
Examples | p. 174 |
Conditional symmetries | p. 179 |
Exercises | p. 182 |
How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariants | p. 184 |
Multiple reduction of variables step by step | p. 184 |
Multiple reduction of variables by using invariants | p. 189 |
Some remarks on group-invariant solutions and their classification | p. 191 |
Exercises | p. 192 |
Symmetries and the separability of partial differential equations | p. 193 |
The problem | p. 193 |
Some remarks on the usual separations of the wave equation | p. 194 |
Hamilton's canonical equations and first integrals in involution | p. 196 |
Quadratic first integrals in involution and the separability of the Hamilton-Jacobi equation and the wave equation | p. 199 |
Exercises | p. 201 |
Contact transformations and contact symmetries of partial differential equations, and how to use them | p. 202 |
The general contact transformation and its infinitesimal generator | p. 202 |
Contact symmetries of partial differential equations and how to find them | p. 204 |
Remarks on how to use contact symmetries for reduction of variables | p. 206 |
Exercises | p. 208 |
Differential equations and symmetries in the language of forms | p. 209 |
Vectors and forms | p. 209 |
Exterior derivatives and Lie derivatives | p. 212 |
Differential equations in the language of forms | p. 213 |
Symmetries of differential equations in the language of forms | p. 215 |
Exercises | p. 219 |
Lie-Backlund transformations | p. 220 |
Why study more general transformations and symmetries? | p. 220 |
Finite order generalizations do not exist | p. 223 |
Lie-Backlund transformations and their infinitesimal generators | p. 225 |
Examples of Lie-Backlund transformations | p. 227 |
Lie-Backlund versus Backlund transformations | p. 229 |
Exercises | p. 231 |
Lie-Backlund symmetries and how to find them | p. 232 |
The basic definitions | p. 232 |
Remarks on the structure of the set of Lie-Backlund symmetries | p. 233 |
How to find Lie-Backlund symmetries: some general remarks | p. 236 |
Examples of Lie-Backlund symmetries | p. 237 |
Recursion operators | p. 242 |
Exercises | p. 245 |
How to use Lie-Backlund symmetries | p. 246 |
Generating solutions by finite symmetry transformations | p. 246 |
Similarity solutions for Lie-Backlund symmetries | p. 248 |
Lie-Backlund symmetries and conservation laws | p. 250 |
Lie-Backlund symmetries and generation methods | p. 251 |
Exercises | p. 252 |
A short guide to the literature | p. 253 |
Solutions to some of the more difficult exercises | p. 255 |
Index | p. 259 |
Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780521366892
ISBN-10: 0521366895
Audience:
Professional
Format:
Paperback
Language:
English
Number Of Pages: 276
Published: 8th October 1990
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.73 x 15.14
x 1.3
Weight (kg): 0.32
Earn 126 Qantas Points
on this Book