TOPOLOGICAL METHODS IN HYDRODYNAMICS
CONTENTS I GROUP AND HAMILTONIAN STRUCTURES OF FLUID DYNAMICS §1 SYMMETRY GROUPS FOR A RIGID BODY AND AN IDEAL ?UID §2 LIE GROUPS, LIE ALGEBRAS, AND ADJOINT REPRESENTATION §3 COADJOINT REPRESENTATION OF A LIE GROUP 3 A DE?NITION OF THE COADJOINT REPRESENTATION 3 B DUAL OF THE SPACE OF PLANE DIVERGENCE-FREE VECTOR ?ELDS 3 C THE LIE ALGEBRA OF DIVERGENCE-FREE VECTOR ?ELDS AND ITS DUAL IN ARBITRARY DIMENSION §4 LEFT-INVARIANT METRICS AND A RIGID BODY FOR AN ARBITRARY GROUP §5 APPLICATIONS TO HYDRODYNAMICS §6 HAMILTONIAN STRUCTURE FOR THE EULER EQUATIONS §7 IDEAL HYDRODYNAMICS ON RIEMANNIAN MANIFOLDS 7 A THE EULER HYDRODYNAMIC EQUATION ON MANIFOLDS 7 B DUAL SPACE TO THE LIE ALGEBRA OF DIVERGENCE-FREE ?ELDS 7 C INERTIA OPERATOR OF AN N-DIMENSIONAL ?UID §8 PROOFS OF THEOREMS ABOUT THE LIE ALGEBRA OF DIVERGENCE-FREE ?ELDS AND ITS DUAL §9 CONSERVATION LAWS IN HIGHER-DIMENSIONAL HYDRODYNAMICS §10 THE GROUP SETTING OF IDEAL MAGNETOHYDRODYNAMICS 10 A EQUATIONS OF MAGNETOHYDRODYNAMICS AND THE KIRCHHOFF EQUATIONS 10 B MAGNETIC EXTENSION OF ANY LIE GROUP 10 C HAMILTONIAN FORMULATION OF THE KIRCHHOFF AND MAGNETOHYDRODYNAMICS EQUATIONS §11 FINITE-DIMENSIONAL APPROXIMATIONS OF THE EULER EQUATION 11 A APPROXIMATIONS BY VORTEX SYSTEMS IN THE PLANE 11 B NONINTEGRABILITY OF FOUR OR MORE POINT VORTICES 11 C HAMILTONIAN VORTEX APPROXIMATIONS IN THREE DIMENSIONS 11 D FINITE-DIMENSIONAL APPROXIMATIONS OF DIFFEOMORPHISM GROUPS §12 THE NAVIER–STOKES EQUATION FROM THE GROUP VIEWPOINT II TOPOLOGY OF STEADY FLUID FLOWS §1 CLASSI?CATION OF THREE-DIMENSIONAL STEADY ?OWS 1 A STATIONARY EULER SOLUTIONS AND BERNOULLI FUNCTIONS 1 B STRUCTURAL THEOREMS §2 VARIATIONAL PRINCIPLES FOR STEADY SOLUTIONS AND APPLICATIONS TO TWO-DIMENSIONAL ?OWS 2 A MINIMIZATION OF THE ENERGY 2 B THE DIRICHLET PROBLEM AND STEADY ?OWS 2 C RELATION OF TWO VARIATIONAL PRINCIPLES 2 D SEMIGROUP VARIATIONAL PRINCIPLE FOR TWO-DIMENSIONAL STEADY ?OWS §3 STABILITY OF STATIONARY POINTS ON LIE ALGEBRAS §4 STABILITY OF PLANAR ?UID ?OWS 4 A STABILITY CRITERIA FOR STEADY ?OWS 4 B WANDERING SOLUTIONS OF THE EULER EQUATION §5 LINEAR AND EXPONENTIAL STRETCHING OF PARTICLES AND RAPIDLY OSCILLATING PERTURBATIONS 5 A THE LINEARIZED AND SHORTENED EULER EQUATIONS 5 B THE ACTION–ANGLE VARIABLES 5 C SPECTRUM OF THE SHORTENED EQUATION 5 D THE SQUIRE THEOREM FOR SHEAR ?OWS 5 E STEADY ?OWS WITH EXPONENTIAL STRETCHING OF PARTICLES 5 F ANALYSIS OF THE LINEARIZED EULER EQUATION 5 G INCONCLUSIVENESS OF THE STABILITY TEST FOR SPACE STEADY ?OWS §6 FEATURES OF HIGHER-DIMENSIONAL STEADY ?OWS 6 A GENERALIZED BELTRAMI ?OWS 6 B STRUCTURE OF FOUR-DIMENSIONAL STEADY ?OWS 6 C TOPOLOGY OF THE VORTICITY FUNCTION 6 D NONEXISTENCE OF SMOOTH STEADY ?OWS AND SHARPNESS OF THE RESTRICTIONS III TOPOLOGICAL PROPERTIES OF MAGNETIC AND VORTICITY FIELDS §1 MINIMAL ENERGY AND HELICITY OF A FROZEN-IN ?ELD 1 A VARIATIONAL PROBLEM FOR MAGNETIC ENERGY 1 B EXTREMAL ?ELDS AND THEIR TOPOLOGY 1 C HELICITY BOUNDS THE ENERGY 1 D HELICITY OF ?ELDS ON MANIFOLDS §2 TOPOLOGICAL OBSTRUCTIONS TO ENERGY RELAXATION 2 A MODEL EXAMPLE: TWO LINKED ?UX TUBES 2 B ENERGY LOWER BOUND FOR NONTRIVIAL LINKING §3 SAKHAROV–ZELDOVICH MINIMIZATION PROBLEM §4 ASYMPTOTIC LINKING NUMBER 4 A ASYMPTOTIC LINKING NUMBER OF A PAIR OF TRAJECTORIES 4 B DIGRESSION ON THE GAUSS FORMULA 4 C ANOTHER DE?NITION OF THE ASYMPTOTIC LINKING NUMBER 4 D LINKING FORMS ON MANIFOLDS §5 ASYMPTOTIC CROSSING NUMBER 5 A ENERGY MINORATION FOR GENERIC VECTOR ?ELDS 5 B ASYMPTOTIC CROSSING NUMBER OF KNOTS AND LINKS 5 C CONFORMAL MODULUS OF A TORUS §6 ENERGY OF A KNOT 6 A ENERGY OF A CHARGED LOOP 6 B GENERALIZATIONS OF THE KNOT ENERGY §7 GENERALIZED HELICITIES AND LINKING NUMBERS 7 A RELATIVE HELICITY 7 B ERGODIC MEANING OF HIGHER-DIMENSIONAL HELICITY INTEGRALS 7 C HIGHER-ORDER LINKING INTEGRALS 7 D CALUGAREANU INVARIANT AND SELF-LINKING NUMBER 7 E HOLOMORPHIC LINKING NUMBER §8 ASYMPTOTIC HOLONOMY AND APPLICATIONS 8 A JONES–WITTEN INVARIANTS FOR VECTOR ?ELDS 8 B INTERPRETATION OF GODBILLON–VEY-TYPE CHARACTERISTIC CLASSES IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS §1 THE LOBACHEVSKY PLANE AND PRELIMINARIES IN DIFFERENTIAL GEOMETRY 1 A THE LOBACHEVSKY PLANE OF AF?NE TRANSFORMATIONS 1 B CURVATURE AND PARALLEL TRANSLATION 1 C BEHAVIOR OF GEODESICS ON CURVED MANIFOLDS 1 D RELATION OF THE COVARIANT AND LIE DERIVATIVES §2 SECTIONAL CURVATURES OF LIE GROUPS EQUIPPED WITH A ONE-SIDED INVARIANT METRIC §3 RIEMANNIAN GEOMETRY OF THE GROUP OF AREA-PRESERVING DIFFEOMORPHISMS OF THE TWO-TORUS 3 A THE CURVATURE TENSOR FOR THE GROUP OF TORUS DIFFEOMORPHISMS 3 B CURVATURE CALCULATIONS §4 DIFFEOMORPHISM GROUPS AND UNRELIABLE FORECASTS 4 A CURVATURES OF VARIOUS DIFFEOMORPHISM GROUPS 4 B UNRELIABILITY OF LONG-TERM WEATHER PREDICTIONS §5 EXTERIOR GEOMETRY OF THE GROUP OF VOLUME-PRESERVING DIFFEOMORPHISMS §6 CONJUGATE POINTS IN DIFFEOMORPHISM GROUPS §7 GETTING AROUND THE ?NITENESS OF THE DIAMETER OF THE GROUP OF VOLUME-PRESERVING DIFFEOMORPHISMS 7 A INTERPLAY BETWEEN THE INTERNAL AND EXTERNAL GEOMETRY OF THE DIFFEOMORPHISM GROUP 7 B DIAMETER OF THE DIFFEOMORPHISM GROUPS 7 C COMPARISON OF THE METRICS AND COMPLETION OF THE GROUP OF DIFFEOMORPHISMS 7 D THE ABSENCE OF THE SHORTEST PATH 7 E DISCRETE ?OWS 7 F OUTLINE OF THE PROOFS 7 G GENERALIZED ?OWS 7 H APPROXIMATION OF ?UID ?OWS BY GENERALIZED ONES 7 I EXISTENCE OF CUT AND CONJUGATE POINTS ON DIFFEOMORPHISM GROUPS §8 IN?NITE DIAMETER OF THE GROUP OF HAMILTONIAN DIFFEOMORPHISMS AND SYMPLECTO-HYDRODYNAMICS 8 A RIGHT-INVARIANT METRICS ON SYMPLECTOMORPHISMS 8 B CALABI INVARIANT 8 C BI-INVARIANT METRICS AND PSEUDOMETRICS ON THE GROUP OF HAMILTONIAN DIFFEOMORPHISMS 8 D BI-INVARIANT INDE?NITE METRIC AND ACTION FUNCTIONAL ON THE GROUP OF VOLUME-PRESERVING DIFFEOMORPHISMS OF A THREE-FOLD V KINEMATIC FAST DYNAMO PROBLEMS §1 DYNAMO AND PARTICLE STRETCHING 1 A FAST AND SLOW KINEMATIC DYNAMOS 1 B NONDISSIPATIVE DYNAMOS ON ARBITRARY MANIFOLDS §2 DISCRETE DYNAMOS IN TWO DIMENSIONS 2 A DYNAMO FROM THE CAT MAP ON A TORUS 2 B HORSESHOES AND MULTIPLE FOLDINGS IN DYNAMO CONSTRUCTIONS 2 C DISSIPATIVE DYNAMOS ON SURFACES 2 D ASYMPTOTIC LEFSCHETZ NUMBER §3 MAIN ANTIDYNAMO THEOREMS 3 A COWLING’S AND ZELDOVICH’S THEOREMS 3 B ANTIDYNAMO THEOREMS FOR TENSOR DENSITIES 3 C DIGRESSION ON THE FOKKER–PLANCK EQUATION 3 D PROOFS OF THE ANTIDYNAMO THEOREMS 3 E DISCRETE VERSIONS OF ANTIDYNAMO THEOREMS §4 THREE-DIMENSIONAL DYNAMO MODELS 4 A “ROPE DYNAMO” MECHANISM 4 B NUMERICAL EVIDENCE OF THE DYNAMO EFFECT 4 C A DISSIPATIVE DYNAMO MODEL ON A THREE-DIMENSIONAL RIEMANNIAN MANIFOLD 4 D GEODESIC ?OWS AND DIFFERENTIAL OPERATIONS ON SURFACES OF CONSTANT NEGATIVE CURVATURE 4 E ENERGY BALANCE AND SINGULARITIES OF THE EULER EQUATION §5 DYNAMO EXPONENTS IN TERMS OF TOPOLOGICAL ENTROPY 5 A TOPOLOGICAL ENTROPY OF DYNAMICAL SYSTEMS 5 B BOUNDS FOR THE EXPONENTS IN NONDISSIPATIVE DYNAMO MODELS 5 C UPPER BOUNDS FOR DISSIPATIVE L1-DYNAMOS VI DYNAMICAL SYSTEMS WITH HYDRODYNAMICAL BACKGROUND §1 THE KORTEWEG–DE VRIES EQUATION AS AN EULER EQUATION 1 A VIRASORO ALGEBRA 1 B THE TRANSLATION ARGUMENT PRINCIPLE AND INTEGRABILITY OF THE HIGH-DIMENSIONAL RIGID BODY 1 C INTEGRABILITY OF THE KDV EQUATION 1 D DIGRESSION ON LIE ALGEBRA COHOMOLOGY AND THE GELFAND–FUCHS COCYCLE §2 EQUATIONS OF GAS DYNAMICS AND COMPRESSIBLE ?UIDS 2 A BAROTROPIC ?UIDS AND GAS DYNAMICS 2 B OTHER CONSERVATIVE ?UID SYSTEMS 2 C IN?NITE CONDUCTIVITY EQUATION §3 K¨AHLER GEOMETRY AND DYNAMICAL SYSTEMS ON THE SPACE OF KNOTS 3 A GEOMETRIC STRUCTURES ON THE SET OF EMBEDDED CURVES 3 B FILAMENT, NONLINEAR SCHR¨ODINGER, AND HEISENBERG CHAIN EQUATIONS 3 C LOOP GROUPS AND THE GENERAL LANDAU–LIFSCHITZ EQUATION §4 SOBOLEV’S EQUATION §5 ELLIPTIC COORDINATES FROM THE HYDRODYNAMICAL VIEWPOINT 5 A CHARGES ON QUADRICS IN THREE DIMENSIONS 5 B CHARGES ON HIGHER-DIMENSIONAL QUADRICS REFERENCES