Elementary Lie Group Analysis and Ordinary Differential Equations

Transcription

1 Elementary Lie Group Analysis and Ordinary Differential Equations Nail H. Ibragimov University of North-West Mmabatho, South Africa JOHN WILEY & SONS Chichester New York Weinheim Brisbane Singapore Toronto

2 Contents Prologue Series Preface Preface xiii xv xvii Part I. Introduction to differential equations 1 1 Model differential equations Introduction Problems from calculus and geometry Integration by quadrature Differential equations for families of curves Minimum surface of revolution Natural phenomena Free fall of a body near the earth Meteoroid A model of rainfall Motion of planets Elementary physics in everyday life Educated farmer Outflow from a funnel Heating and air conditioning Electrical instruments. Equation of van der Pol Mechanical vibrations Collapse of driving shafts Ecology Radioactive decay Population growth Predator and prey Competing species 23 Problems 24

3 vi CONTENTS 2 Elementary methods of integration First-order equations Linear equations. Superposition of solutions Separation of variables. A nonlinear superposition Notation of exterior differential calculus Exact equations Integrating factor Homogeneous equations Equations of the type y' = filx+hj+m) The Riccati equation The Bernoulli equation Higher-order equations Integration of the equation y^ = f(x) Equations F(y, y',..., y (n) ) = 0. Reduction of order General and intermediate integrals F(x, y,y',..., y (n) ) = 0 with F a total derivative The equations y( n > = /(y ( "- 1} ) and y( n > = /(y ( "- 2) ) Equations F(x,t/W,y( fc+1 ),...,2/("))= Linear equations. Method of variation of parameters Linear equations with constant coefficients Systems of first-order linear equations 42 Problems^ 44 3 General properties of solutions Existence and uniqueness theorems Classical solutions. Cauchy's problem The method of successive approximations Systems of first-order equations Higher order equations Analytic solutions. The method of majorants Systems of analytic equations of the first order Example: The exponential map Equations of the first order not solved for y' Equations algebraic in y and y' Equations x = f{y') and y = f(y'). Parametric solutions Lagrange's equation Clairaut's equation. The envelope of the general integral as a singular solution Global behavior of solutions Singularities of first-order equations Equations in the complex domain. Movable and fixed singular points 62 Problems 64

4 CONTENTS 4 Partial differential equations of the first order Notation Equations of the first order and classical solutions Linear, quasi-linear and nonlinear equations Integration of linear equations First integrals of systems of ordinary differential equations Homogeneous linear partial differential equations Non-homogeneous "equations Quasi-linear equations Laplace's method Extension of Laplace's method to many variables Reduction to a homogeneous linear equation Integral surfaces as loci of characteristic curves Nonlinear equations Complete, general and singular integrals Completely integrable systems. The Lagrange-Charpit method Solution of Cauchy's problem via complete integrals Monge's theory of characteristics Example: Light rays as visible characteristics Cauchy's method Hamilton's equations - characteristics of the Hamilton-Jacobi equation Systems of homogeneous linear equations Basic notions Complete systems Integration of complete systems 93 Problems 95 vii Part II: Fundamentals of Lie group analysis 97 5 Gateway to modern group analysis: Historical survey Appearance of groups in the theory of equations Sophus Lie and symmetry analysis of differential equations His life story Symmetry groups, Lie algebras and integration of ordinary differential equations Group classification of differential equations Linearization of second-order equations Nonlinear superposition Tangent transformations Contact transformations and their applications Infinitesimal contact transformations 107

5 viii CONTENTS Higher order tangent transformations: Lie and Backlund's discussion Modern theory of Lie-Backlund transformation groups Applied group analysis Symmetry and conservation laws Restoration of group analysis in the 1960s Ill Invariant and partially invariant solutions Ill Symmetry of fluids New trends Invariance principle in boundary-value problems "Hidden" symmetries, in particular non-local symmetries Approximate symmetries Group theoretic modelling Miscellany 118 Problems Preliminaries on transformations and groups Transformations in elementary mathematics Algebra: Solution of equations Geometry: Similarity and calculation of areas Transformations in M n Changes of coordinates and point transformations 122 x Definition of a group Subgroups A,display of transformation groups Transformations of the straight line Groups in the plane Motions, conformal mappings and other groups in IR Different types of groups: Continuous, discontinuous and mixed. Path curves. Global and local groups 132 Problems Infinitesimal transformations and local groups One-parameter groups Notation and assumptions Definition of a local group Representation of local groups in a canonical parameter Infinitesimal transformations The Lie equations. The exponential map Determination of a canonical parameter Invariants Canonical and semi-canonical variables Three methods of construction of one-parameter groups with known infinitesimal generators 145

6 CONTENTS ix Table of usual one-parameter groups in the plane, their generators, invariants and canonical variables Invariant equations The infinitesimal test ~ Invariant representation of invariant manifolds Examples on Theorem Introduction to Lie algebras Definition of Lie algebras of operators Notation. Table of commutators Subalgebra and ideal Quotient algebra Derived algebras Solvable Lie algebras Isomorphic and similar Lie algebras Non-isomorphic structures and non-similar realizations in the plane of 1, 2, and 3-dimensional Lie algebras Multi-parameter groups: Outline of basic notions Definition Composition via one-parameter groups Basis of invariants Regular and singular invariant manifolds Approximate transformation groups Motivation Notation One-parameter approximate transformation groups Approximate group generator A preparatory lemma Lie equations in the first order of precision The approximate exponential map Examples on the exponential map 179 Problems Calculus of differential algebra Main variables and total derivatives The universal space A of modern group analysis Differential functions and the space A Successive derivatives of differential functions One independent and one differential variable. Faa de Bruno's formula Extended point transformation groups. Contact transformations Transformations of the plane One independent and several differential variables Point transformations involving many variables Properties of extended generators 192

7 CONTENTS Differential invariants. Invariant differentiation Change of derivatives under differential substitutions Infinitesimal contact transformations Irreducible contact transformation groups in the plane Operators and identities in A The Euler-Lagrange operator. Test for a total derivative Lie-Backlund operators Operators N* associated with Lie-Backlund operators The fundamental identity The frame of differential equations 204 Problems 206 Symmetry of differential equations Notation and assumptions Determination of infinitesimal symmetries Two definitions of a symmetry group Determining equations Intrinsic symmetries of mathematical models Samples for solution of determining equations 213 9:3.1 Ordinary differential equations: Theorems on the maximum number of symmetries and examples Extended generators and calculation of symmetries of partial differential equations with two independent variables Solution of the determining equations for a system Determination of equations admitting a given group Contact symmetries Invariant solutions Lie's theory of invariant solutions Inherited symmetries of equations for invariant solutions Approximate symmetries of equations with a small parameter Definition of an approximate symmetry group Determining equations Calculation of infinitesimal approximate symmetries On discontinuous and non-local symmetries Symmetry and conservation laws Principle of least action. Euler-Lagrange equations Differential algebraic proof of Noether's theorem Examples from classical mechanics Derivation of Kepler's laws from symmetries Conservation laws of relativistic mechanics 243 Problems 247

8 CONTENTS xi 10 Invariants of algebraic and differential equations Invariants of algebraic equations Preliminaries Tschirnhausen's transformation. An approach to the Galois group Infinitesimal method Linear ordinary differential equations Preliminaries Infinitesimal method Nonlinear ordinary differential equations Linear partial differential equations The Laplace invariants The Ovsyannikov invariants The Maxwell equations 262 Problems 263 Part III. Basic integration methods First-order equations and systems Integration by using an infinitesimal symmetry Lie's integrating factor Method of canonical variables Group interpretation of variation of parameters Systems admitting nonlinear superposition Superposition formulas for nonlinear systems Main theorem. The Vessiot-Guldberg-Lie algebra Examples Projective interpretation of the Riccati equation Linearizable Riccati equations Decoupling and integration of systems using Vessiot-Guldberg-Lie algebras Application: An invariant solution of nonlinear equations modelling laser systems 279 Problems Second-order equations Reduction of order Canonical variables Invariant differentiation Integration by means of two symmetries Consecutive integration Canonical forms of two-dimensional Lie algebras Lie's integration algorithm 289

9 xii CONTENTS A sample for implementation of the algorithm Application to the non-homogeneous linear equation Lie's linearization test Main theorem Examples Integration using approximate symmetries 299 Problems Higher-order equations Third-order equations Equations admitting a solvable Lie algebra Equations admitting a non-solvable Lie algebra Group theoretical background of Euler's method for linear equations with constant coefficients Symmetries and invariant solutions The case of multiple roots 305 Problems 306 Answers 307 Notes 325 Index 343