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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 | |