| Preface |
| Introduction / 1： |
| Ordinary differential equations / I： |
| Point transformations and their generators / 2： |
| One-parameter groups of point transformations and their infinitesimal generators / 2.1： |
| Transformation laws and normal forms of generators / 2.2： |
| Extensions of transformations and their generators / 2.3： |
| Multiple-parameter groups of transformations and their generators / 2.4： |
| Exercises / 2.5： |
| Lie point symmetries of ordinary differential equations: the basic definitions and properties / 3： |
| The definition of a symmetry: first formulation / 3.1： |
| Ordinary differential equations and linear partial differential equations of first order / 3.2： |
| The definition of a symmetry: second formulation / 3.3： |
| Summary / 3.4： |
| How to find the Lie point symmetries of an ordinary differential equation / 3.5： |
| Remarks on the general procedure / 4.1： |
| The atypical case: first order differential equations / 4.2： |
| Second order differential equations / 4.3： |
| Higher order differential equations. The general nth order linear equation / 4.4： |
| How to use Lie point symmetries: differential equations with one symmetry / 4.5： |
| First order differential equations / 5.1： |
| Higher order differential equations / 5.2： |
| Some basic properties of Lie algebras / 5.3： |
| The generators of multiple-parameter groups and their Lie algebras / 6.1： |
| Examples of Lie algebras / 6.2： |
| Subgroups and subalgebras / 6.3： |
| Realizations of Lie algebras. Invariants and differential invariants / 6.4： |
| Nth order differential equations with multiple-parameter symmetry groups: an outlook / 6.5： |
| How to use Lie point symmetries: second order differential equations admitting a G[subscript 2] / 6.6： |
| A classification of the possible subcases, and ways one might proceed / 7.1： |
| The first integration strategy: normal forms of generators in the space of variables / 7.2： |
| The second integration strategy: normal forms of generators in the space of first integrals / 7.3： |
| Summary: Recipe for the integration of second order differential equations admitting a group G[subscript 2] / 7.4： |
| Examples / 7.5： |
| Second order differential equations admitting more than two Lie point symmetries / 7.6： |
| The problem: groups that do not contain a G[subscript 2] / 8.1： |
| How to solve differential equations that admit a G[subscript 3] IX / 8.2： |
| Example / 8.3： |
| Higher order differential equations admitting more than one Lie point symmetry / 8.4： |
| The problem: some general remarks / 9.1： |
| First integration strategy: normal forms of generators in the space(s) of variables / 9.2： |
| Second integration strategy: normal forms of generators in the space of first integrals. Lie's theorem / 9.3： |
| Third integration strategy: differential invariants / 9.4： |
| Systems of second order differential equations / 9.5： |
| The corresponding linear partial differential equation of first order and the symmetry conditions / 10.1： |
| Example: the Kepler problem / 10.2： |
| Systems possessing a Lagrangian: symmetries and conservation laws / 10.3： |
| Symmetries more general than Lie point symmetries / 10.4： |
| Why generalize point transformations and symmetries? / 11.1： |
| How to generalize point transformations and symmetries / 11.2： |
| Contact transformations / 11.3： |
| How to find and use contact symmetries of an ordinary differential equation / 11.4： |
| Dynamical symmetries: the basic definitions and properties / 11.5： |
| What is a dynamical symmetry? / 12.1： |
| Examples of dynamical symmetries / 12.2： |
| The structure of the set of dynamical symmetries / 12.3： |
| How to find and use dynamical symmetries for systems possessing a Lagrangian / 12.4： |
| Dynamical symmetries and conservation laws / 13.1： |
| Example: L = (x[superscript 2] + y[superscript 2])/2 - a(2y[superscript 3] + x[superscript 2]y), a [characters not producible] 0 / 13.2： |
| Example: geodesics of a Riemannian space - Killing vectors and Killing tensors / 13.3： |
| Systems of first order differential equations with a fundamental system of solutions / 13.5： |
| The problem / 14.1： |
| The answer / 14.2： |
| Systems with a fundamental system of solutions and linear systems / 14.3： |
| Partial differential equations / 14.5： |
| Lie point transformations and symmetries / 15： |
| The definition of a symmetry / 15.1： |
| How to determine the point symmetries of partial differential equations / 15.4： |
| How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry transformations / 16.1： |
| The structure of the set of symmetry generators / 17.1： |
| What can symmetry transformations be expected to achieve? / 17.2： |
| Generating solutions by finite symmetry transformations / 17.3： |
| Generating solutions (of linear differential equations) by applying the generators / 17.4： |
| How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variables / 17.5： |
| Similarity variables and how to find them / 18.1： |
| Conditional symmetries / 18.3： |
| How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariants / 18.5： |
| Multiple reduction of variables step by step / 19.1： |
| Multiple reduction of variables by using invariants / 19.2： |
| Some remarks on group-invariant solutions and their classification / 19.3： |
| Symmetries and the separability of partial differential equations / 19.4： |
| Some remarks on the usual separations of the wave equation / 20.1： |
| Hamilton's canonical equations and first integrals in involution / 20.3： |
| Quadratic first integrals in involution and the separability of the Hamilton-Jacobi equation and the wave equation / 20.4： |
| Contact transformations and contact symmetries of partial differential equations, and how to use them / 20.5： |
| The general contact transformation and its infinitesimal generator / 21.1： |
| Contact symmetries of partial differential equations and how to find them / 21.2： |
| Remarks on how to use contact symmetries for reduction of variables / 21.3： |
| Differential equations and symmetries in the language of forms / 21.4： |
| Vectors and forms / 22.1： |
| Exterior derivatives and Lie derivatives / 22.2： |
| Differential equations in the language of forms / 22.3： |
| Symmetries of differential equations in the language of forms / 22.4： |
| Lie-Backlund transformations / 22.5： |
| Why study more general transformations and symmetries? / 23.1： |
| Finite order generalizations do not exist / 23.2： |
| Lie-Backlund transformations and their infinitesimal generators / 23.3： |
| Examples of Lie-Backlund transformations / 23.4： |
| Lie-Backlund versus Backlund transformations / 23.5： |
| Lie-Backlund symmetries and how to find them / 23.6： |
| The basic definitions / 24.1： |
| Remarks on the structure of the set of Lie-Backlund symmetries / 24.2： |
| How to find Lie-Backlund symmetries: some general remarks / 24.3： |
| Examples of Lie-Backlund symmetries / 24.4： |
| Recursion operators / 24.5： |
| How to use Lie-Backlund symmetries / 24.6： |
| Similarity solutions for Lie-Backlund symmetries / 25.1： |
| Lie-Backlund symmetries and conservation laws / 25.3： |
| Lie-Backlund symmetries and generation methods / 25.4： |
| A short guide to the literature / 25.5： |
| Solutions to some of the more difficult exercises / Appendix B： |
| Index |
| Preface |
| Introduction / 1： |
| Ordinary differential equations / I： |
図書
