HAMILTONIAN DYNAMICAL SYSTEMS AND APPLICATIONS
CONTENTS SOME ASPECTS OF ?NITE-DIMENSIONAL HAMILTONIAN DYNAMICS SYMPLECTIC STRUCTURE INVARIANT FORM OF THE HAMILTONIAN EQUATIONS HAMILTONIAN EQUATIONS THE POISSON BRACKET LIOUVILLE THEOREM ON COMPLETELY INTEGRABLE SYSTEMS A PENDULUM WITH RAPIDLY OSCILLATING SUSPENTION POINT ANTI-INTEGRABLE LIMIT THE STANDARD MAP PROOF OF THE AUBRY THEOREM SOME REMARKS SEPARATRIX SPLITTING POINCARɒS OBSERVATION THE POINCARÉ INTEGRAL PROOF OF THEOREM 5 STANDARD EXAMPLE REFERENCES FOUR LECTURES ON THE N-BODY PROBLEM THE POINCARɖBIRKHOFF–CONLEY TWIST MAP OF THE ANNULUS FOR THE PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM THE KEPLER PROBLEM AS AN OSCILLATOR THE RESTRICTED PROBLEM IN THE LUNAR CASE HILL’S SOLUTIONS THE ANNULUS TWIST MAP THE ARNOLD–HERMAN STABILITY THEOREM FOR THE SPATIAL (1 + N)-BODY PROBLEM THE SECULAR HAMILTONIAN HERMAN’S NORMAL FORM THEOREM AND HOW TO USE IT A STABILITY THEOREM HERMAN’S DEGENERACY MINIMAL ACTION AND MARCHAL’S THEOREM CENTRAL CON?GURATIONS AND THEIR HOMOGRAPHIC MOTIONS VARIATIONAL CHARACTERIZATIONS OF LAGRANGE’S EQUILATERAL SOLUTIONS MARCHAL’S THEOREM MINIMIZATION UNDER SYMMETRY CONSTRAINTS GLOBAL CONTINUATION VIA MINIMIZATION BIFURCATIONS FROM THE LAGRANGE EQUILATERAL RELATIVE EQUILIBRIUM FROM THE EQUILATERAL TRIANGLE TO THE EIGHT FROM THE SQUARE TO THE HIP-HOP THE AVATARS OF THE REGULAR N-GON RELATIVE EQUILIBRIUM: EIGHTS, CHAINS AND GENERALIZED HIP-HOPS AVERAGING METHOD AND ADIABATIC INVARIANTS INTRODUCTION ADIABATIC INVARIANCE IN ONE-FREQUENCY SYSTEMS ON ADIABATIC INVARIANCE IN MULTI-FREQUENCY SYSTEMS TRANSFORMATION THEORY OF HAMILTONIAN PDE AND THE PROBLEM OF WATER WAVES WALTER CRAIG HAMILTONIAN SYSTEMS PARTIAL DIFFERENTIAL EQUATIONS AS HAMILTONIAN SYSTEMS THE PROBLEM OF WATER WAVES THE DIRICHLET–NEUMANN OPERATOR PERTURBATION THEORY THE CALCULUS OF TRANSFORMATIONS THREE THEOREMS ON PERTURBED KDV KDV EQUATION INTEGRABILITY OF (KDV) NORMAL FORMS FOR (KDV) KAM-THEORY AVERAGING: HAMILTONIAN PERTURBATIONS AVERAGING: CASE OF NON-HAMILTONIAN PERTURBATIONS DETERMINISTIC PERTURBATIONS RANDOM PERTURBATIONS GROUPS AND TOPOLOGY IN THE EULER HYDRODYNAMICS AND KDV EULER EQUATIONS AND GEODESICS THE EULER HYDRODYNAMICS EQUATION GEODESICS ON LIE GROUPS GEODESIC DESCRIPTION FOR VARIOUS EQUATIONS TOPOLOGY OF STEADY ?OWS ARNOLD’S CLASSI?CATION OF STEADY ?UID ?OWS VARIATIONAL PRINCIPLES FOR STEADY ?OWS EULER EQUATIONS AND INTEGRABLE SYSTEMS HAMILTONIAN REFORMULATION OF THE EULER EQUATIONS THE VIRASORO ALGEBRA AND THE KDV EQUATION EQUATIONS–RELATIVES AND CONSERVATION LAWS IN?NITE DIMENSIONAL DYNAMICAL SYSTEMS AND THE NAVIER–STOKES EQUATION FIRST LECTURE: IN?NITE DIMENSIONAL DYNAMICAL SYSTEMS SECOND LECTURE: INVARIANT MANIFOLDS FOR PARTIAL DIFFERENTIAL EQUATIONS ON UNBOUNDED DOMAINS THIRD LECTURE: AN INTRODUCTION TO THE NAVIER–STOKES EQUATIONS FOURTH LECTURE: THE LONG-TIME ASYMPTOTICS OF SOLUTIONS OF THE TWO-DIMENSIONAL NAVIER–STOKES EQUATION CONCLUSIONS HAMILTONIAN SYSTEMS AND OPTIMAL CONTROL FIRST LECTURE FIRST ORDER CONDITIONS SECOND LECTURE SECOND VARIATION THIRD LECTURE CURVES IN THE LAGRANGE GRASSMANNIANS CURVATURE-TYPE INVARIANTS FOURTH LECTURE KAM THEORY WITH APPLICATIONS TO HAMILTONIAN PARTIAL DIFFERENTIAL EQUATIONS BRIEF HISTORY AND BASIC IDEAS OF KAM THEORY DERIVATION OF THE LINEARIZED EQUATIONS SOLUTIONS OF THE LINEARIZED EQUATIONS APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS IN HIGHER DIMENSIONS FOUR LECTURES ON KAM FOR THE NON-LINEAR SCHRÖDINGER EQUATION THE NON-LINEAR SCHRÖDINGER EQUATION AN ?-DIMENSIONAL HAMILTONIAN SYSTEM THE TOPOLOGY ACTION-ANGLE VARIABLES STATEMENT OF THE RESULT KAM-TORI CONSEQUENCES OF THEOREM 1 NOTATION THE HOMOLOGICAL EQUATION NORMAL FORM HAMILTONIANS THE KAM-ITERATION THE COMPONENTS OF THE HOMOLOGICAL EQUATION SMALL DIVISORS AND THE SECOND MELNIKOV CONDITION BLOCKS LIPSCHITZ DOMAINS TÖPLITZ AT ?(D = 2) TÖPLITZ–LIPSCHITZ MATRICES (D = 2) ESTIMATES OF SMALL DIVISORS A BASIC ESTIMATE THE SECOND MELNIKOV CONDITION (D = 2) FUNCTIONS WITH THE TÖPLITZ–LIPSHITZ PROPERTY (D = 2) TÖPLITZ STRUCTURE OF THE HESSIAN TÖPLITZ–LIPSCHITZ MATRICES L ×L ?GL(2,R) FUNCTIONS WITH THE TÖPLITZ–LIPSCHITZ PROPERTY A SHORT REMARK ON THE PROOF OF THEOREM 11 A BIRKHOFF NORMAL FORM THEOREM FOR SOME SEMILINEAR PDES BIRKHOFF’S THEOREM IN ?NITE DIMENSIONS STATEMENT PROOF THE CASE OF PDES HAMILTONIAN FORMULATION OF THE WAVE EQUATION EXTENSION OF BIRKHOFF’S THEOREM TO PDES: HEURISTIC IDEAS A BIRKHOFF NORMAL FORM THEOREM FOR SEMILINEAR PDES MAPS WITH LOCALIZED COEF?CIENTS AND THEIR PROPERTIES STATEMENT OF THE NORMAL FORM THEOREM AND ITS CONSEQUENCES APPLICATION TO THE NONLINEAR WAVE EQUATION DISCUSSION PROOFS PROOF OF THE PROPERTIES OF FUNCTIONS WITH LOCALIZED COEF?CIENTS AND OF ITS DYNAMICAL CONSEQUENCES PROOF OF PROPOSITION 42 ON THE VERI?CATION OF THE PROPERTY OF LOCALIZATION OF COEF?CIENTS PROOF OF THEOREM 44 ON THE NONRESONANCE CONDITION NORMAL FORM OF HOLOMORPHIC DYNAMICAL SYSTEMS DE?NITIONS AND EXAMPLES VECTOR ?ELDS AND DIFFERENTIAL EQUATIONS NORMAL FORMS OF VECTOR ?ELDS EXAMPLES ABOUT LINEARIZATION EXAMPLES ABOUT NONLINEARIZABLE VECTOR ?ELDS HOLOMORPHIC NORMALIZATION THEOREM OF AD BRJUNO THEOREMS OF J VEY SINGULAR COMPLETE INTEGRABILITY–MAIN RESULT HOW TO RECOVER BRJUNO’S AND VEY’S THEOREMS FROM THEOREM 236 SKETCH OF THE PROOF PROOF OF MAIN THEOREM 236 BOUNDS FOR THE COHOMOLOGICAL EQUATIONS ITERATION SCHEME PROOF OF THE THEOREM MISCELLANEOUS RESULTS NORMAL FORMS AGAIN KAM THEORY POISSON STRUCTURES GEOMETRIC APPROACHES TO THE PROBLEM OF INSTABILITY IN HAMILTONIAN SYSTEMS AN INFORMAL PRESENTATION AMADEU DELSHAMS, MARIAN GIDEA, RAFAEL DE LA LLAVE, AND TERE M SEARA TWO TYPES OF GEOMETRIC PROGRAMS EXPOSITION OF THE ARNOLD EXAMPLE THE OBSTRUCTION PROPERTY SOME ?NAL REMARKS ON THE EXAMPLE IN [ARN64] RETURN TO A NORMALLY HYPERBOLIC MANIFOLD THE TWO DYNAMICS APPROACH THE BASICS OF THE MECHANISM OF RETURN TO A NORMALLY HYPERBOLIC INVARIANT MANIFOLD THE SCATTERING MAP THE SCATTERING MAP AND HOMOCLINIC INTERSECTIONS OF SUBMANIFOLDS MONODROMY OF THE SCATTERING MAP SMOOTHNESS AND SMOOTH DEPENDENCE ON PARAMETERS GEOMETRIC PROPERTIES OF THE SCATTERING MAP CALCULATION OF THE SCATTERING MAP THE LARGE GAP MODEL GENERATION OF INTERSECTIONS MELNIKOV THEORY FOR NORMALLY HYPERBOLIC MANIFOLDS COMPUTATION OF THE SCATTERING MAP THE AVERAGING METHOD RESONANT AVERAGING REPEATED AVERAGING INVARIANT OBJECTS GENERATED BY RESONANCES: SECONDARY TORI, LOWER DIMENSIONAL TORI HETEROCLINIC INTERSECTIONS BETWEEN THE INVARIANT OBJECTS GENERATED BY RESONANCES THE METHOD OF CORRECTLY ALIGNED WINDOWS THE LARGE GAP MODEL: THE METHOD OF CORRECTLY ALIGNED WINDOWS THE LARGE GAP MODEL IN HIGHER DIMENSIONS INSTABILITY CAUSED BY NORMALLY HYPERBOLIC LAMINATIONS MODELS WITH TWO TIME SCALES: GEODESIC ?OWS, BILLIARDS WITH MOVING BOUNDARIES, LITTLEWOOD PROBLEMS APPENDIX VARIATIONAL METHODS FOR THE PROBLEM OF ARNOLD DIFFUSION INTRODUCTION TO MATHER THEORY EXISTENCE OF HOMOCLINIC ORBITS PSEUDO-CONNECTING ORBIT SET ˜ C?,µ,? EXISTENCE OF HETEROCLINIC ORBITS CONSTRUCTION OF GLOBAL CONNECTING ORBITS APPLICATION TO A PRIORI UNSTABLE SYSTEMS THE CALCULUS OF VARIATIONS AND THE FORCED PENDULUM PERIODIC SOLUTIONS HETEROCLINIC SOLUTIONS MULTITRANSITION SOLUTIONS: THE SIMPLEST CASE MULTITRANSITION SOLUTIONS: GENERAL CASE THE TIP OF THE ICEBERG VARIATIONAL METHODS FOR HAMILTONIAN PDES FINITE DIMENSIONS: RESONANT CENTER THEOREMS THE VARIATIONAL LYAPUNOV–SCHMIDT REDUCTION IN?NITE DIMENSIONS THE BIFURCATION EQUATION THE SMALL DIVISOR PROBLEM THE (Q1)-EQUATION A VARIATIONAL PRINCIPLE ON A CANTOR SET FORCED VIBRATIONS SPECTRAL GAPS OF POTENTIALS IN WEIGHTED SOBOLEV SPACES RESULTS REDUCTION GAP ESTIMATES COEF?CIENT ESTIMATES MODI?ED WEIGHTS ON THE WELL-POSEDNESS OF THE PERIODIC KDV EQUATION IN HIGH REGULARITY CLASSES BIRKHOFF COORDINATES REGULARITY