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| Preface | |
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| Introduction | |
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| Ordinary differential equations | |
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| Point transformations and their generators | |
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| One-parameter groups of point transformations and their infinitesimal generators | |
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| Transformation laws and normal forms of generators | |
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| Extensions of transformations and their generators | |
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| Multiple-parameter groups of transformations and their generators | |
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| Exercises | |
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| Lie point symmetries of ordinary differential equations: the basic definitions and properties | |
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| The definition of a symmetry: first formulation | |
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| Ordinary differential equations and linear partial differential equations of first order | |
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| The definition of a symmetry: second formulation | |
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| Summary | |
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| Exercises | |
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| How to find the Lie point symmetries of an ordinary differential equation | |
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| Remarks on the general procedure | |
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| The atypical case: first order differential equations | |
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| Second order differential equations | |
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| Higher order differential equations. The general nth order linear equation | |
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| Exercises | |
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| How to use Lie point symmetries: differential equations with one symmetry | |
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| First order differential equations | |
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| Higher order differential equations | |
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| Exercises | |
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| Some basic properties of Lie algebras | |
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| The generators of multiple-parameter groups and their Lie algebras | |
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| Examples of Lie algebras | |
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| Subgroups and subalgebras | |
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| Realizations of Lie algebras. Invariants and differential invariants | |
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| Nth order differential equations with multiple-parameter symmetry groups: an outlook | |
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| Exercises | |
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| How to use Lie point symmetries: second order differential equations admitting a G[subscript 2] | |
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| A classification of the possible subcases, and ways one might proceed | |
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| The first integration strategy: normal forms of generators in the space of variables | |
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| The second integration strategy: normal forms of generators in the space of first integrals | |
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| Summary: Recipe for the integration of second order differential equations admitting a group G[subscript 2] | |
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| Examples | |
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| Exercises | |
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| Second order differential equations admitting more than two Lie point symmetries | |
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| The problem: groups that do not contain a G[subscript 2] | |
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| How to solve differential equations that admit a G[subscript 3] IX | |
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| Example | |
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| Exercises | |
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| Higher order differential equations admitting more than one Lie point symmetry | |
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| The problem: some general remarks | |
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| First integration strategy: normal forms of generators in the space(s) of variables | |
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| Second integration strategy: normal forms of generators in the space of first integrals. Lie's theorem | |
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| Third integration strategy: differential invariants | |
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| Examples | |
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| Exercises | |
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| Systems of second order differential equations | |
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| The corresponding linear partial differential equation of first order and the symmetry conditions | |
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| Example: the Kepler problem | |
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| Systems possessing a Lagrangian: symmetries and conservation laws | |
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| Exercises | |
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| Symmetries more general than Lie point symmetries | |
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| Why generalize point transformations and symmetries? | |
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| How to generalize point transformations and symmetries | |
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| Contact transformations | |
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| How to find and use contact symmetries of an ordinary differential equation | |
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| Exercises | |
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| Dynamical symmetries: the basic definitions and properties | |
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| What is a dynamical symmetry? | |
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| Examples of dynamical symmetries | |
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| The structure of the set of dynamical symmetries | |
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| Exercises | |
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| How to find and use dynamical symmetries for systems possessing a Lagrangian | |
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| Dynamical symmetries and conservation laws | |
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| Example: L = (x[superscript 2] + y[superscript 2])/2 - a(2y[superscript 3] + x[superscript 2]y), a [characters not producible] 0 | |
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| Example: the Kepler problem | |
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| Example: geodesics of a Riemannian space - Killing vectors and Killing tensors | |
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| Exercises | |
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| Systems of first order differential equations with a fundamental system of solutions | |
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| The problem | |
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| The answer | |
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| Examples | |
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| Systems with a fundamental system of solutions and linear systems | |
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| Exercises | |
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| Partial differential equations | |
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| Lie point transformations and symmetries | |
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| Introduction | |
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| Point transformations and their generators | |
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| The definition of a symmetry | |
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| Exercises | |
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| How to determine the point symmetries of partial differential equations | |
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| First order differential equations | |
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| Second order differential equations | |
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| Exercises | |
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| How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry transformations | |
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| The structure of the set of symmetry generators | |
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| What can symmetry transformations be expected to achieve? | |
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| Generating solutions by finite symmetry transformations | |
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| Generating solutions (of linear differential equations) by applying the generators | |
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| Exercises | |
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| How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variables | |
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| The problem | |
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| Similarity variables and how to find them | |
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| Examples | |
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| Conditional symmetries | |
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| Exercises | |
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| How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariants | |
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| Multiple reduction of variables step by step | |
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| Multiple reduction of variables by using invariants | |
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| Some remarks on group-invariant solutions and their classification | |
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| Exercises | |
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| Symmetries and the separability of partial differential equations | |
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| The problem | |
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| Some remarks on the usual separations of the wave equation | |
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| Hamilton's canonical equations and first integrals in involution | |
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| Quadratic first integrals in involution and the separability of the Hamilton-Jacobi equation and the wave equation | |
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| Exercises | |
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| Contact transformations and contact symmetries of partial differential equations, and how to use them | |
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| The general contact transformation and its infinitesimal generator | |
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| Contact symmetries of partial differential equations and how to find them | |
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| Remarks on how to use contact symmetries for reduction of variables | |
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| Exercises | |
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| Differential equations and symmetries in the language of forms | |
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| Vectors and forms | |
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| Exterior derivatives and Lie derivatives | |
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| Differential equations in the language of forms | |
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| Symmetries of differential equations in the language of forms | |
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| Exercises | |
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| Lie-Backlund transformations | |
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| Why study more general transformations and symmetries? | |
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| Finite order generalizations do not exist | |
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| Lie-Backlund transformations and their infinitesimal generators | |
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| Examples of Lie-Backlund transformations | |
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| Lie-Backlund versus Backlund transformations | |
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| Exercises | |
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| Lie-Backlund symmetries and how to find them | |
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| The basic definitions | |
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| Remarks on the structure of the set of Lie-Backlund symmetries | |
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| How to find Lie-Backlund symmetries: some general remarks | |
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| Examples of Lie-Backlund symmetries | |
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| Recursion operators | |
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| Exercises | |
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| How to use Lie-Backlund symmetries | |
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| Generating solutions by finite symmetry transformations | |
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| Similarity solutions for Lie-Backlund symmetries | |
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| Lie-Backlund symmetries and conservation laws | |
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| Lie-Backlund symmetries and generation methods | |
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| Exercises | |
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| A short guide to the literature | |
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| Solutions to some of the more difficult exercises | |
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| Index | |