Linear Differential Transformations of the Second Order

Transcription

1 Linear Differential Transformations of the Second Order In: Otakar Borůvka (author); Felix M. Arscott (translator): Linear Differential Transformations of the Second Order. (English). London: The English Universities Press, Ltd., pp. IX XVI. Persistent URL: Terms of use: The English Universities Press, Ltd. Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Survey PREPARATION PHASE THEORY THE TRANSFORMATION PROBLEM TRANSFORMATION THEORY DISPERSION THEORY GENERAL TRANSFORMATION THEORY Central dispersions General dispersions General transformations Complete transformations RECENT DEVELOPMENTS IN TRANSFORMATION THEORY I FOUNDATIONS OF THE THEORY General properties of ordinary linear homogeneous differential equations of the second order 3 1 Introduction 1.1 Preliminaries 1.2 The Wronskian determinant 1.3 Bases 1.4 Integral curves 1.5 Kinematic interpretation of integrals 1.6 Types of differential equations (q) 1.7 The Schwarzian derivative 1.8 Projective property of the Schwarzian derivative 1.9 Associated differential equations ix

3 x 2 Elementary properties of integrals of the differential equation (q) Relative positions of zeros of an integral and its derivative Ratios of integrals and their derivatives The ordering theorems 12 pi fa r*i q( a \ 2.4 The (Riemann) integrals -TT-T> -T^T T da in the neighbourhood of Jx 0 y 2 {a) Jx 0 y 2 (a) a singular point Application to the associated equation Basis functions 15 3 Conjugate numbers The concept of conjugate numbers Classification of differential equations (q) with respect to conjugate numbers Properties of differential equations (q) with tc-conjugate numbers Fundamental numbers Fundamental integrals and fundamental sequences General and special equations of finite type Relations between conjugate numbers of different kinds Equations with proper fundamental numbers Singular bases Differential equations (q) with 1-conjugate numbers Differential equations (q) with conjugate numbers of all four kinds Bilinear relations between integrals of the differential equation (q) 26 4 Centro-affine differential geometry of plane curves Representation of plane curves Centro-affine representatives of a plane curve and its representations Centro-affine invariants of plane curves Regular curves Centro-affine curvature Application of the above theory to integral curves of the differential equation (q) 32 Phase theory of ordinary linear homogeneous differential equations of the second order 34 5 Polar coordinates of bases Introduction Amplitudes First phases of a basis Boundedness of a first phase Continuity property of a first phase First phases of the differential equation (q) Phase functions Second phases of a basis Boundedness of a second phase 40

4 xi 5.10 Continuity property of a second phase Connection of a second phase with the associated differential equation Second phases of the differential equation (q) Integrals of the differential equation (q) and their derivatives expressed in polar coordinates Ordering relations between first and second phases of the same basis Some consequences Explicit connection between first and second phases of the same basis Phases of different bases of the J differential equation (q) 46 "*i pi g(a) da, h(a) da in the neighbourhood X0 of singular points 47 6 Polar functions The concept of polar functions General polar form of the carrier q Determination of the carrier from a polar function Radon functions Normalized polar functions Normalized polar functions of the first kind Determination of the carrier from a first normalized polar function Normalized polar functions of the second kind Determination of the carrier from a second normalized polar function Normalized polar functions of the third kind Determination of the carrier from a third normalized polar function Some applications of polar functions 62 JXQ 7 Local and boundary properties of phases Unique determination of a phase from the Cauchy initial conditions Boundary values of phases Normalized boundary values of phases Singular phases Properties of singular phases Boundary values of null phases Boundary values of other phases ' * 7.8 Normal phases '2 7.9 Structure of the set of singular normal phases of a differential equation (Sv Structure of a phase bundle Structure of a phase bunch The mapping of p into the phase OL P Relations between zeros and boundary values of normal phases Boundary characteristics of normal phases Determination of normal phases with given characteristic triple ^ 7.16 Determination of the type and kind of the differential equation (q) ^y means of boundary values of a phase of (q) 2

5 xii 7.17 Properties of second phases Relations between the boundary values of a first and second phase of the same basis 84 8 Elementary phases Introduction Properties of equations with elementary phases Properties of integrals, and their derivatives, of differential equations (q) with elementary phases Determination of all carriers with elementary first phases Equations with elementary phases, defined over ( oo, oo) Power of the set of elementary carriers Generalization of the concept of elementary phases 92 9 Relations between first phases of two differential equations (q), (Q) Introduction Linked phases Associated numbers Characteristic triples of two differential equations Similar phases Existence of similar phases Algebraic structure of the set of phases of oscillatory differential equations (q) in the interval ( oo, oo) The phase group ( The equivalence relation Q The fundamental subgroup The subgroup $ of elementary phases 104 II DISPERSION THEORY 107 Theory of central dispersions The transformation problem Historical background Formulation of the transformation problem Introduction to the theory of central dispersions Some preliminaries Definition of the central dispersions Central dispersions of oscillatory differential equations (q) Relations between central dispersions Algebraic structure of the set of central dispersions 116

6 xiii 13 Properties of central dispersions Monotonicity and continuity The functional equation of the central dispersions Derivatives of central dispersions Higher derivatives The connection between central dispersions and the transformation problem Relations between derivatives of the central dispersions and the values of the carrier a Relations between central dispersions and phases Representation of central dispersions and their derivatives by means of phases Structure of the Abel functional equations Representation of the central dispersions by normalized polar functions Differential equations of the central dispersions Solutions of the Abel functional equations with unknown phase functions a,jff Consequences of the above results 133 Special problems of central dispersions Extension of solutions of a differential equation (q) and their derivatives Extension of solutions of the differential equation (q) Extension of derivatives of solutions of the differential equation (q) Differential equations with the same central dispersions of the first kind Integral strips Statement of the problem Properties of the integral strips of the set (Q</>) A sufficient condition for two differential equations to have the same fundamental dispersion Ratios of elements of an integral strip Relations between carriers with the same fundamental dispersion <j> Explicit formula for carriers with the same fundamental dispersion <f> Explicit formulae for elementary carriers Relations between first phases of differential equations with the same fundamental dispersion os Power of the set Q<f> Differential equations with coincident central dispersions of the K-th and (K + l)-th kinds (K = 1,3) 149 I. Theory of F-carriers Characteristic properties Domain of definition of F-carriers Elementary carriers Kinematic properties of F-carriers 151

7 xiv II. Theory of K-carriers Characteristic properties of J?-carriers Further properties of J?-carriers Connection between i^-carriers and Radon curves Connection between R- and F-carriers Kinematic properties of K-carriers Bunch curves and Radon curves Fundamental concepts Determination of the carriers of the bunch curves Centro-affine length of arcs of bunch curves Finite equations of bunch curves Radon curves 162 Theory of general dispersions Introduction Dispersions of the #c-th kind; K = 1, 2, 3, General dispersions Linear mapping of the integral spaces of the differential equations (qx (Q) on each other Fundamental concepts Composition of mappings Mapping of an integral space into itself Determination of the linear mappings from the first phases Phase bases of composed mappings Determination of a mapping from arbitrary phases Normalized linear mappings Canonical phase bases General dispersions of the differential equations (q), (Q) Fundamental numbers and fundamental intervals of the differential equation (q) The concept of general dispersion Properties of general dispersions Determining elements of general dispersions Integration of the differential equation (Qq) Connection between general dispersions and the transformation problem Embedding of the general dispersions in the phase group Dispersions of the K-th kind; K = 1, 2, 3, Introduction Determination of dispersions of the /c-th kind, K = 1, 2, 3, Determination of the central dispersions of the /<r-th kind; #c == 1, 2, 3, The group of dispersions of the first kind of the differential equation (q) 186

8 xv 21.5 Group property of dispersions of the first kind Representation of the dispersion group 189 2L7 The group of dispersions of the second kind of the differential equation (q) The semigroupoid of general dispersions of the differential equations (q), (Q) 192 HI GENERAL TRANSFORMATION THEORY 195 General transformations Establishment of the special form of the transformation formula A theorem on transformations of second order differential equations Introduction of the differential equation (Qq) Transformation properties of solutions of the differential equation (Qq) Relations between solutions of the differential equations (Qq), (qq)? (qq), (QQ) Reciprocal transformations of integrals of the differential equations (q), (Q) Transformations of the derivatives of integrals of the differential equations (q), (Q) Relations between solutions of the differential equation (Qq) and first phases of the differential equations (q), (Q) Reciprocal transformations of first and second phases of the differential equations (q)? (Q) Existence and uniqueness problems for solutions of the differential equation (Qq) The existence and uniqueness theorem for solutions of the differential equation (Qq) Transformations of given integrals of the differential equations (q)? (Q) into each other Physical application of general transformation theory Straight line motion in physical space Harmonic motion 214 Complete transformations Existence and generality of complete transformations Formulation of the problem Preliminary The existence problem for complete solutions of the differential equation (Qq) The multiplicity of the complete solutions of the differential equation (Qq) 217

9 xvi 27 Structure of the set of complete solutions of the differential equation (Qq) Preliminary Relations between complete solutions of the differential equation (Qq) The structure of the set of complete solutions of the differential equation (Qq) in the case of differential equations (q), (Q) of finite type m? m > Canonical forms of the differential equation (q) 223 IV RECENT DEVELOPMENTS OF TRANSFORMATION THEORY An abstract algebraic model for the transformation theory of Jacobian oscillatory differential equations Structure of the group of second order regular matrices over the real number field An abstract phase group Linear vector spaces Quasinorms Kummer transformations of bases Kummer transformations of elements Abstract dispersions Realization of the abstract model by means of analytic transformation theory A survey of recent results in transformation theory General dispersions Dispersions of the 1st and 2nd kinds Central dispersions Generalizations 242 Bibliography 243 Supplementary Bibliography 248 Author and Subject Index 251