Differential Equations Their Solution Using Symmetries

ISBN-10: 0521366895
ISBN-13: 9780521366892
Edition: 1989
List price: $57.00 Buy it from $14.81
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Description: In many branches of physics, mathematics, and engineering, solving a problem means solving a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The emphasis in this text is  More...

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Book details

List price: $57.00
Copyright year: 1989
Publisher: Cambridge University Press
Publication date: 1/26/1990
Binding: Paperback
Pages: 276
Size: 6.00" wide x 9.50" long x 0.75" tall
Weight: 0.990
Language: English

In many branches of physics, mathematics, and engineering, solving a problem means solving a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The emphasis in this text is on how to find and use the symmetries; this is supported by many examples and more than 100 exercises. This book will form an introduction accessible to beginning graduate students in physics, applied mathematics, and engineering. Advanced graduate students and researchers in these disciplines will find the book a valuable reference.

Malcolm MacCallum is Director of the Heilbronn Institute at the University of Bristol and is President of the International Society on General Relativity and Gravitation.

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

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