| 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 |
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