GEOMETRICAL THEORY OF DYNAMICAL SYSTEMS AND FLUID FLOWS
CONTENTS I MATHEMATICAL BASES 1 MANIFOLDS, FLOWS, LIE GROUPS AND LIE ALGEBRAS 1 1 DYNAMICAL SYSTEMS 1 2 MANIFOLDS AND DI?EOMORPHISMS 1 3 FLOWS AND VECTOR FIELDS A STEADY ?OW AND ITS VELOCITY ?ELD TANGENT VECTOR AND DI?ERENTIAL OPERATOR TANGENT SPACE TIME-DEPENDENT (UNSTEADY) VELOCITY ?ELD 1 4 DYNAMICAL TRAJECTORY FIBER BUNDLE (TANGENT BUNDLE) LAGRANGIAN AND HAMILTONIAN LEGENDRE TRANSFORMATION 1 5 DI?ERENTIAL AND INNER PRODUCT COVECTOR (1-FORM) INNER (SCALAR) PRODUCT 1 6 MAPPING OF VECTORS AND COVECTORS PUSH-FORWARD TRANSFORMATION PULL-BACK TRANSFORMATION COORDINATE TRANSFORMATION 1 7 LIE GROUP AND INVARIANT VECTOR FIELDS 1 8 LIE ALGEBRA AND LIE DERIVATIVE LIE ALGEBRA, ADJOINT OPERATOR AND LIE BRACKET AN EXAMPLE OF THE ROTATION GROUP SO(3) LIE DERIVATIVE AND LAGRANGE DERIVATIVE 1 9 DI?EOMORPHISMS OF A CIRCLE S1 1 10 TRANSFORMATION OF TENSORS AND INVARIANCE 1 10 1 TRANSFORMATIONS OF VECTORS AND METRIC TENSORS 1 10 2 COVARIANT TENSORS 1 10 3 MIXED TENSORS 1 10 4 CONTRAVARIANT TENSORS 2 1 FIRST FUNDAMENTAL FORM 2 2 SECOND FUNDAMENTAL FORM 2 3 GAUSS’S SURFACE EQUATION AND AN INDUCED CONNECTION 2 4 GAUSS–MAINARDI–CODAZZI EQUATION AND INTEGRABILITY 2 5 GAUSSIAN CURVATURE OF A SURFACE RIEMANN TENSORS GAUSSIAN CURVATURE GEODESIC CURVATURE AND NORMAL CURVATURE PRINCIPAL CURVATURES 2 6 GEODESIC EQUATION 2 7 STRUCTURE EQUATIONS IN DI?ERENTIAL FORMS SMOOTH SURFACES IN R3 AND INTEGRABILITY STRUCTURE EQUATIONS GEODESIC EQUATION 2 8 GAUSS SPHERICAL MAP 2 9 GAUSS–BONNET THEOREM I 2 10 GAUSS–BONNET THEOREM II 2 11 UNIQUENESS: FIRST AND SECOND FUNDAMENTAL TENSORS 3 RIEMANNIAN GEOMETRY 3 1 TANGENT SPACE TANGENT VECTORS AND INNER PRODUCT RIEMANNIAN METRIC EXAMPLES OF METRIC TENSOR 3 2 COVARIANT DERIVATIVE (CONNECTION) DE?NITION TIME-DEPENDENT CASE 3 3 RIEMANNIAN CONNECTION CHRISTO?EL SYMBOL 3 4 COVARIANT DERIVATIVE ALONG A CURVE DERIVATIVE ALONG A PARAMETERIZED CURVE PARALLEL TRANSLATION DYNAMICAL SYSTEM OF AN INVARIANT METRIC 3 5 STRUCTURE EQUATIONS STRUCTURE EQUATIONS AND CONNECTION FORMS TWO-DIMENSIONAL SURFACE M EXAMPLE: POINCAR´E SURFACE (I) 3 6 GEODESIC EQUATION LOCAL COORDINATE REPRESENTATION GROUP-THEORETIC REPRESENTATION EXAMPLE: POINCAR´E SURFACE (II) 3 7 COVARIANT DERIVATIVE AND PARALLEL TRANSLATION PARALLEL TRANSLATION AGAIN COVARIANT DERIVATIVE AGAIN A FORMULA OF COVARIANT DERIVATIVE 3 8 ARC-LENGTH 3 9 CURVATURE TENSOR AND CURVATURE TRANSFORMATION CURVATURE TRANSFORMATION CURVATURE TENSOR SECTIONAL CURVATURE RICCI TENSOR AND SCALAR CURVATURE 3 10 JACOBI EQUATION 3 10 1 DERIVATION 3 10 2 INITIAL BEHAVIOR OF JACOBI ?ELD 3 10 3 TIME-DEPENDENT PROBLEM 3 10 4 TWO-DIMENSIONAL PROBLEM 3 10 5 ISOTROPIC SPACE 3 11 DI?ERENTIATION OF TENSORS 3 11 1 LIE DERIVATIVES OF 1-FORM AND METRIC TENSOR 3 11 2 RIEMANNIAN CONNECTION ? 3 11 3 COVARIANT DERIVATIVE OF TENSORS 3 12 KILLING FIELDS 3 12 1 KILLING VECTOR ?ELD X 3 12 2 ISOMETRY 3 12 3 POSITIVE CURVATURE AND SIMPLI?ED JACOBI EQUATION 3 12 4 CONSERVATION OF ?X, T?ALONG ?(S) 3 12 5 KILLING TENSOR ?ELD 3 13 INDUCED CONNECTION AND SECOND FUNDAMENTAL FORM II DYNAMICAL SYSTEMS 4 FREE ROTATION OF A RIGID BODY 4 1 PHYSICAL BACKGROUND FREE ROTATION AND EULER’S TOP INTEGRALS OF MOTION LIE–POISSON BRACKET AND HAMILTON’S EQUATION 4 2 TRANSFORMATIONS (ROTATIONS) BY SO(3) TRANSFORMATION OF REFERENCE FRAMES RIGHT-INVARIANCE AND LEFT-INVARIANCE 4 3 COMMUTATOR AND RIEMANNIAN METRIC 4 4 GEODESIC EQUATION LEFT-INVARIANT DYNAMICS RIGHT-INVARIANT DYNAMICS 4 5 BI-INVARIANT RIEMANNIAN METRICES SO(3) IS COMPACT AD-INVARIANCE AND BI-INVARIANT METRICES CONNECTION AND CURVATURE TENSOR 4 6 ROTATING TOP AS A BI-INVARIANT SYSTEM A SPHERICAL TOP (EUCLIDEAN METRIC) AN ASYMMETRICAL TOP (RIEMANNIAN METRIC) SYMMETRICAL TOP AND ITS STABILITY STABILITY AND INSTABILITY OF AN ASYMMETRICAL TOP SUPPLEMENTARY NOTES TO §4 6 3 5 WATER WAVES AND KDV EQUATION 5 1 PHYSICAL BACKGROUND: LONG WAVES IN SHALLOW WATER 5 2 SIMPLE DI?EOMORPHIC FLOW COMMUTATOR AND METRIC OF D(S1) GEODESIC EQUATION ON D(S1) SECTIONAL CURVATURES ON D(S1) 5 3 CENTRAL EXTENSION OF D(S1) 5 4 KDV EQUATION AS A GEODESIC EQUATION ON ˆD(S1) 5 5 KILLING FIELD OF KDV EQUATION KILLING EQUATION ISOMETRY GROUP INTEGRAL INVARIANT CONJUGATE POINT 5 6 SECTIONAL CURVATURES OF KDV SYSTEM 6 HAMILTONIAN SYSTEMS: CHAOS, INTEGRABILITY AND PHASE TRANSITION 6 1 A DYNAMICAL SYSTEM WITH SELF-INTERACTION HAMILTONIAN AND METRIC TENSOR JACOBI EQUATION METRIC AND COVARIANT DERIVATIVE 6 2 TWO DEGREES OF FREEDOM POTENTIALS 6 3 H´ENON–HEILES MODEL AND CHAOS CONVENTIONAL METHOD EVIDENCE OF CHAOS IN A GEOMETRICAL ASPECT 6 4 GEOMETRY AND CHAOS 6 5 INVARIANTS IN A GENERALIZED MODEL KILLING VECTOR ?ELD ANOTHER INTEGRABLE CASE 6 6 TOPOLOGICAL SIGNATURE OF PHASE TRANSITIONS MORSE FUNCTION AND EULER INDEX SIGNATURES OF PHASE TRANSITION TOPOLOGICAL CHANGE IN THE MEAN-?ELD XY MODEL III FLOWS OF IDEAL FLUIDS 7 GAUGE PRINCIPLE AND VARIATIONAL FORMULATION 7 1 INTRODUCTION: FLUID FLOWS AND FIELD THEORY 7 2 LAGRANGIANS AND VARIATIONAL PRINCIPLE GALILEI-INVARIANT LAGRANGIAN HAMILTON’S VARIATIONAL FORMULATIONS LAGRANGE’S EQUATION 7 3 CONCEPTUAL SCENARIO OF THE GAUGE PRINCIPLE 7 4 GLOBAL GAUGE TRANSFORMATION 7 5 LOCAL GAUGE TRANSFORMATION COVARIANT DERIVATIVE LAGRANGIAN 7 6 SYMMETRIES OF FLOW FIELDS TRANSLATIONAL TRANSFORMATION ROTATIONAL TRANSFORMATION RELATIVE DISPLACEMENT 7 7 LAWS OF TRANSLATIONAL TRANSFORMATION LOCAL GALILEI TRANSFORMATION DETERMINATION OF GAUGE ?ELD A IRROTATIONAL ?ELDS ?(X) AND U(X) 7 8 FLUID FLOWS AS MATERIAL MOTION LAGRANGIAN PARTICLE REPRESENTATION LAGRANGE DERIVATIVE AND LIE DERIVATIVE KINEMATICAL CONSTRAINT 7 9 GAUGE-FIELD LAGRANGIAN LA (TRANSLATIONAL SYMMETRY) A POSSIBLE FORM LAGRANGIAN OF BACKGROUND THERMODYNAMIC STATE 7 10 HAMILTON’S PRINCIPLE FOR POTENTIAL FLOWS 7 10 1 LAGRANGIAN 7 10 2 MATERIAL VARIATIONS: IRROTATIONAL AND ISENTROPIC 7 10 3 CONSTRAINTS FOR VARIATIONS 7 10 4 ACTION PRINCIPLE FOR LP 7 11 ROTATIONAL TRANSFORMATIONS 7 11 1 ORTHOGONAL TRANSFORMATION OF VELOCITY 7 11 2 IN?NITESIMAL TRANSFORMATIONS 7 12 GAUGE TRANSFORMATION (ROTATION) 7 12 1 LOCAL GAUGE TRANSFORMATION 7 12 2 COVARIANT DERIVATIVE ?T 7 12 3 GAUGE PRINCIPLE 7 12 4 TRANSFORMATION LAW OF THE GAUGE ?ELD ? 7 13 GAUGE-FIELD LAGRANGIAN LB (ROTATIONAL SYMMETRY) 7 14 BIOT–SAVART’S LAW 7 14 1 VECTOR POTENTIAL OF MASS ?UX 7 14 2 VORTICITY AS A GAUGE ?ELD 7 15 HAMILTON’S PRINCIPLE FOR AN IDEAL FLUID (ROTATIONAL FLOWS) 7 15 1 CONSTITUTIVE CONDITIONS 7 15 2 LAGRANGIAN AND ITS VARIATIONS 7 15 3 MATERIAL VARIATION: ROTATIONAL AND ISENTROPIC 7 15 4 EULER’S EQUATION OF MOTION 7 15 5 CONSERVATIONS OF MOMENTUM AND ENERGY 7 15 6 NOETHER’S THEOREM FOR ROTATIONS 7 16 LOCAL SYMMETRIES IN A-SPACE 7 16 1 EQUATION OF MOTION IN A-SPACE 7 16 2 VORTICITY EQUATION AND LOCAL ROTATION SYMMETRY 7 16 3 VORTICITY EQUATION IN THE X-SPACE 7 16 4 KELVIN’S CIRCULATION THEOREM 7 16 5 LAGRANGIAN OF THE GAUGE ?ELD 7 17 CONCLUSIONS 8 VOLUME-PRESERVING FLOWS OF AN IDEAL FLUID 8 1 FUNDAMENTAL CONCEPTS VOLUME-PRESERVING DI?EOMORPHISMS RIGHT-INVARIANT ?ELDS 8 2 BASIC TOOLS COMMUTATOR DIVERGENCE-FREE CONNECTION COADJOINT ACTION AD? FORMULAS IN R3 SPACE 8 3 GEODESIC EQUATION 8 4 JACOBI EQUATION AND FROZEN FIELD 8 5 INTERPRETATION OF RIEMANNIAN CURVATURE OF FLUID FLOWS FLAT CONNECTION PRESSURE GRADIENT AS AN AGENT YIELDING CURVATURE INSTABILITY IN LAGRANGIAN PARTICLE SENSE TIME EVOLUTION OF JACOBI ?ELD STRETCHING OF LINE-ELEMENTS 8 6 FLOWS ON A CUBIC SPACE (FOURIER REPRESENTATION) 8 7 LAGRANGIAN INSTABILITY OF PARALLEL SHEAR FLOWS NEGATIVE SECTIONAL CURVATURES STABILITY OF A PLANE COUETTE ?OW OTHER PARALLEL SHEAR ?OWS 8 8 STEADY FLOWS AND BELTRAMI FLOWS STEADY ?OWS A BELTRAMI ?OW ABC ?OW W + DF 9 MOTION OF VORTEX FILAMENTS 9 1 A VORTEX FILAMENT 9 2 FILAMENT EQUATION 9 3 BASIC PROPERTIES LEFT-INVARIANCE AND RIGHT-INVARIANCE LANDAU–LIFSHITZ EQUATION METRIC AND LOOP ALGEBRA 9 4 GEOMETRICAL FORMULATION AND GEODESIC EQUATION 9 5 VORTEX FILAMENTS AS A BI-INVARIANT SYSTEM CIRCULAR VORTEX ?LAMENTS GENERAL VORTEX ?LAMENTS INTEGRAL INVARIANTS 9 6 KILLING FIELDS ON VORTEX FILAMENTS A RECTILINEAR VORTEX A CIRCULAR VORTEX A HELICAL VORTEX 9 7 SECTIONAL CURVATURE AND GEODESIC STABILITY KILLING ?ELDS GENERAL TANGENT ?ELD X 9 8 CENTRAL EXTENSION OF THE ALGEBRA OF FILAMENT MOTION IV GEOMETRY OF INTEGRABLE SYSTEMS 10 GEOMETRIC INTERPRETATIONS OF SINE–GORDON EQUATION 10 1 PSEUDOSPHERE: A GEOMETRIC DERIVATION OF SG 10 2 BIANCHI–LIE TRANSFORMATION 10 3 B¨ACKLUND TRANSFORMATION OF SG EQUATION 11 INTEGRABLE SURFACES: RIEMANNIAN GEOMETRY AND GROUP THEORY 11 1 BASIC IDEAS 11 2 PSEUDOSPHERICAL SURFACES: SG, KDV, MKDV, SHG 11 3 SPHERICAL SURFACES: NLS, SG, NSM 11 3 1 NONLINEAR SCHR¨ODINGER EQUATION 11 3 2 SINE–GORDON EQUATION REVISITED 11 3 3 NONLINEAR SIGMA MODEL AND SG EQUATION 11 3 4 SPHERICAL AND PSEUDOSPHERICAL SURFACES 11 4 B¨ACKLUND TRANSFORMATIONS REVISITED 11 4 1 A B¨ACKLUND TRANSFORMATION 11 4 2 SELF-B¨ACKLUND TRANSFORMATION 11 5 IMMERSION OF INTEGRABLE SURFACES ON LIE GROUPS 11 5 1 A SURFACE ?2 IN R3 11 5 2 SURFACES ON LIE GROUPS AND LIE ALGEBRAS 11 5 3 NONLINEAR SCHR¨ODINGER SURFACES 11 6 MAPPING OF INTEGRABLE SYSTEMS TO SPHERICAL SURFACES