Differential equations : their solution using symmetries /

Differential equations : their solution using symmetries /

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Bibliographic Details
Author / Creator:Stephani, Hans
Imprint:Cambridge [England] ; New York : Cambridge University Press, 1989.
Description:xii, 260 p. : ill. ; 24 cm.
Language:English
Subject:
Differential equations -- Numerical solutions
Symmetry
Differential equations -- Numerical solutions.
Symmetry.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/1101758
Hidden Bibliographic Details
Other authors / contributors:MacCallum, M. A. H.
ISBN:0521355311
0521366895 (pbk.)
Notes:Includes bibliographical references (p. [253]-254).
Table of Contents:
  • Preface
  • 1. Introduction
  • I. Ordinary differential equations
  • 2. Point transformations and their generators
  • 2.1. One-parameter groups of point transformations and their infinitesimal generators
  • 2.2. Transformation laws and normal forms of generators
  • 2.3. Extensions of transformations and their generators
  • 2.4. Multiple-parameter groups of transformations and their generators
  • 2.5. Exercises
  • 3. Lie point symmetries of ordinary differential equations: the basic definitions and properties
  • 3.1. The definition of a symmetry: first formulation
  • 3.2. Ordinary differential equations and linear partial differential equations of first order
  • 3.3. The definition of a symmetry: second formulation
  • 3.4. Summary
  • 3.5. Exercises
  • 4. How to find the Lie point symmetries of an ordinary differential equation
  • 4.1. Remarks on the general procedure
  • 4.2. The atypical case: first order differential equations
  • 4.3. Second order differential equations
  • 4.4. Higher order differential equations. The general nth order linear equation
  • 4.5. Exercises
  • 5. How to use Lie point symmetries: differential equations with one symmetry
  • 5.1. First order differential equations
  • 5.2. Higher order differential equations
  • 5.3. Exercises
  • 6. Some basic properties of Lie algebras
  • 6.1. The generators of multiple-parameter groups and their Lie algebras
  • 6.2. Examples of Lie algebras
  • 6.3. Subgroups and subalgebras
  • 6.4. Realizations of Lie algebras. Invariants and differential invariants
  • 6.5. Nth order differential equations with multiple-parameter symmetry groups: an outlook
  • 6.6. Exercises
  • 7. How to use Lie point symmetries: second order differential equations admitting a G[subscript 2]
  • 7.1. A classification of the possible subcases, and ways one might proceed
  • 7.2. The first integration strategy: normal forms of generators in the space of variables
  • 7.3. The second integration strategy: normal forms of generators in the space of first integrals
  • 7.4. Summary: Recipe for the integration of second order differential equations admitting a group G[subscript 2]
  • 7.5. Examples
  • 7.6. Exercises
  • 8. Second order differential equations admitting more than two Lie point symmetries
  • 8.1. The problem: groups that do not contain a G[subscript 2]
  • 8.2. How to solve differential equations that admit a G[subscript 3] IX
  • 8.3. Example
  • 8.4. Exercises
  • 9. Higher order differential equations admitting more than one Lie point symmetry
  • 9.1. The problem: some general remarks
  • 9.2. First integration strategy: normal forms of generators in the space(s) of variables
  • 9.3. Second integration strategy: normal forms of generators in the space of first integrals. Lie's theorem
  • 9.4. Third integration strategy: differential invariants
  • 9.5. Examples
  • 9.6. Exercises
  • 10. Systems of second order differential equations
  • 10.1. The corresponding linear partial differential equation of first order and the symmetry conditions
  • 10.2. Example: the Kepler problem
  • 10.3. Systems possessing a Lagrangian: symmetries and conservation laws
  • 10.4. Exercises
  • 11. Symmetries more general than Lie point symmetries
  • 11.1. Why generalize point transformations and symmetries?
  • 11.2. How to generalize point transformations and symmetries
  • 11.3. Contact transformations
  • 11.4. How to find and use contact symmetries of an ordinary differential equation
  • 11.5. Exercises
  • 12. Dynamical symmetries: the basic definitions and properties
  • 12.1. What is a dynamical symmetry?
  • 12.2. Examples of dynamical symmetries
  • 12.3. The structure of the set of dynamical symmetries
  • 12.4. Exercises
  • 13. How to find and use dynamical symmetries for systems possessing a Lagrangian
  • 13.1. Dynamical symmetries and conservation laws
  • 13.2. Example: L = (x[superscript 2] + y[superscript 2])/2 - a(2y[superscript 3] + x[superscript 2]y), a [characters not producible] 0
  • 13.3. Example: the Kepler problem
  • 13.4. Example: geodesics of a Riemannian space - Killing vectors and Killing tensors
  • 13.5. Exercises
  • 14. Systems of first order differential equations with a fundamental system of solutions
  • 14.1. The problem
  • 14.2. The answer
  • 14.3. Examples
  • 14.4. Systems with a fundamental system of solutions and linear systems
  • 14.5. Exercises
  • II. Partial differential equations
  • 15. Lie point transformations and symmetries
  • 15.1. Introduction
  • 15.2. Point transformations and their generators
  • 15.3. The definition of a symmetry
  • 15.4. Exercises
  • 16. How to determine the point symmetries of partial differential equations
  • 16.1. First order differential equations
  • 16.2. Second order differential equations
  • 16.3. Exercises
  • 17. How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry transformations
  • 17.1. The structure of the set of symmetry generators
  • 17.2. What can symmetry transformations be expected to achieve?
  • 17.3. Generating solutions by finite symmetry transformations
  • 17.4. Generating solutions (of linear differential equations) by applying the generators
  • 17.5. Exercises
  • 18. How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variables
  • 18.1. The problem
  • 18.2. Similarity variables and how to find them
  • 18.3. Examples
  • 18.4. Conditional symmetries
  • 18.5. Exercises
  • 19. How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariants
  • 19.1. Multiple reduction of variables step by step
  • 19.2. Multiple reduction of variables by using invariants
  • 19.3. Some remarks on group-invariant solutions and their classification
  • 19.4. Exercises
  • 20. Symmetries and the separability of partial differential equations
  • 20.1. The problem
  • 20.2. Some remarks on the usual separations of the wave equation
  • 20.3. Hamilton's canonical equations and first integrals in involution
  • 20.4. Quadratic first integrals in involution and the separability of the Hamilton-Jacobi equation and the wave equation
  • 20.5. Exercises
  • 21. Contact transformations and contact symmetries of partial differential equations, and how to use them
  • 21.1. The general contact transformation and its infinitesimal generator
  • 21.2. Contact symmetries of partial differential equations and how to find them
  • 21.3. Remarks on how to use contact symmetries for reduction of variables
  • 21.4. Exercises
  • 22. Differential equations and symmetries in the language of forms
  • 22.1. Vectors and forms
  • 22.2. Exterior derivatives and Lie derivatives
  • 22.3. Differential equations in the language of forms
  • 22.4. Symmetries of differential equations in the language of forms
  • 22.5. Exercises
  • 23. Lie-Backlund transformations
  • 23.1. Why study more general transformations and symmetries?
  • 23.2. Finite order generalizations do not exist
  • 23.3. Lie-Backlund transformations and their infinitesimal generators
  • 23.4. Examples of Lie-Backlund transformations
  • 23.5. Lie-Backlund versus Backlund transformations
  • 23.6. Exercises
  • 24. Lie-Backlund symmetries and how to find them
  • 24.1. The basic definitions
  • 24.2. Remarks on the structure of the set of Lie-Backlund symmetries
  • 24.3. How to find Lie-Backlund symmetries: some general remarks
  • 24.4. Examples of Lie-Backlund symmetries
  • 24.5. Recursion operators
  • 24.6. Exercises
  • 25. How to use Lie-Backlund symmetries
  • 25.1. Generating solutions by finite symmetry transformations
  • 25.2. Similarity solutions for Lie-Backlund symmetries
  • 25.3. Lie-Backlund symmetries and conservation laws
  • 25.4. Lie-Backlund symmetries and generation methods
  • 25.5. Exercises
  • Appendix A. A short guide to the literature
  • Appendix B. Solutions to some of the more difficult exercises
  • Index

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