Hamiltonian Systems with Three or More Degrees of Freedom (Nato Science Series C:, 533)

A survey of current knowledge about Hamiltonian systems with three or more degrees of freedom and related topics. The Hamiltonian systems appearing in most of the applications are non-integrable. Hence methods to prove non-integrability results are presented and the different meaning attributed to non-integrability are discussed. For systems near an integrable one, it can be shown that, under suitable conditions, some parts of the integrable structure, most of the invariant tori, survive. Many of the papers discuss near-integrable systems. From a topological point of view, some singularities must appear in different problems, either caustics, geodesics, moving wavefronts, etc. This is also related to singularities in the projections of invariant objects, and can be used as a signature of these objects. Hyperbolic dynamics appear as a source on unpredictable behaviour and several mechanisms of hyperbolicity are presented. The destruction of tori leads to Aubrey-Mather objects, and this is touched on for a related class of systems. Examples without periodic orbits are constructed, against a classical conjecture. Other topics concern higher dimensional systems, either finite (networks and localised vibrations on them) or infinite, like the quasiperiodic Schrödinger operator or nonlinear hyperbolic PDE displaying quasiperiodic solutions. Most of the applications presented concern celestial mechanics problems, like the asteroid problem, the design of spacecraft orbits, and methods to compute periodic solutions. Front Matter....Pages i-xxiv Front Matter....Pages 1-1 Inflection Points, Extatic Points and Curve Shortening....Pages 3-10 Topologically Necessary Singularities on Moving Wavefronts and Caustics....Pages 11-12 Heteroclinic Chains of Skew Product Hamiltonian Systems....Pages 13-25 Order and Chaos in 3-D Systems....Pages 26-38 Splitting of Separatrices in Hamiltonian Systems and Symplectic Maps....Pages 39-54 On The Discrete One-Dimensional Quasi-Periodic Schrödinger Equation and Other Smooth Quasi-Periodic Skew Products....Pages 55-61 Lindstedt Series and Kolmogorov Theorem....Pages 62-71 A Classical Self-Contained Proof of Kolmogorov’s Theorem on Invariant Tori....Pages 72-89 Dynamical Stability in Lagrangian Systems....Pages 90-114 The Origin of Chaotic Behaviour in the Kirkwood Gaps....Pages 115-125 Examples of Compact Hypersurfaces in R 2P , 2P ≥ 6, With No Periodic Orbits....Pages 126-126 Hamiltonian Systems with Three Degrees of Freedom and Hydrodynamics....Pages 127-133 Introduction to Frequency Map Analysis....Pages 134-150 Lindstedt Series for Lower Dimensional Tori....Pages 151-167 Arnold Diffusion; a Compendium of Remarks and Questions....Pages 168-183 Old and New Applications of Kam Theory....Pages 184-192 On Adiabatic Invariance in Two-Frequency Systems....Pages 193-212 The Method of Rational Approximations: Theory and Applications....Pages 213-222 Dynamical Systems Methods for Space Missions on a Vicinity of Collinear Libration Points....Pages 223-241 A Mechanism of Ergodicity in Standard-Like Maps....Pages 242-243 Front Matter....Pages 1-1 Continuous Averaging in Hamiltonian Systems....Pages 244-253 Phase Space Geometry and Dynamics Associated with the 1:2:2 Resonance....Pages 254-269 From Singular Point Analysis to Rigorous Results on Integrability: a Dream of S. Kowalevskaya....Pages 270-282 Front Matter....Pages 283-283 Time Singularities for Polynomial Hamiltonians with Analytic Time Dependence....Pages 285-289 Numerical Study of Turbulence in N-Body Hamiltonian Systems with Long Range Force: I. Relaxation....Pages 290-294 Phase Space Structures in 3 and 4 Degrees of Freedom: Application to Chemical Reactions....Pages 295-299 Modulated Diffusion for Symplectic Maps....Pages 300-304 On The Jeans-Landau-Teller Approximation for Adiabatic Invariants....Pages 305-309 Periodic Orbits and Quantum Mechanics of Molecular Hamiltonian Systems....Pages 310-313 Chaos in Atom-Surface Collisions....Pages 314-317 On The Non-Integrability of the Mixmaster Universe Model....Pages 318-323 Branching of Solutions as Obstruction to the Existence of a Meromorphic Integral in Many-Dimensional Systems....Pages 324-329 The Energy Surfaces of the Kovalevskaya-Top....Pages 330-334 Filling Rates for Linear Flow on the Torus: Recent Progress and Applications....Pages 335-339 Invariant Spectra of Orbits in Multidimensional Symplectic Maps....Pages 340-344 On the Convergence of Formal Series Containing Small Divisors....Pages 345-349 Systems with an Invariant Measure on Lie Groups....Pages 350-356 Stochasticity of the 2/1 Asteroidal Resonance: a Symplectic Mapping Approach....Pages 357-361 On Perturbed Oscillators in1–1–1 Resonance: Critical Inclination in the 3D Hénon-Heiles Potential....Pages 362-366 Splitting of Separatrices for (Fast) Quasiperiodic Forcing....Pages 367-371 Front Matter....Pages 283-283 A Possible Mechanism for the Kam Tori Breakdown....Pages 372-376 On the Dynamics Near Resonant Equilibria....Pages 377-385 Exponentially Small Estimates for Kam Theorem Near an Elliptic Equilibrium Point....Pages 386-390 Fast Diffusion and Universality Near Intersecting Resonances....Pages 391-397 Quasi-Periodic Motion of a Rigid Body Under Weak Forces....Pages 398-402 Center and Center-(Un)Stable Manifolds of Elliptic-Hyperbolic Fixed Points of 4 D -Symplectic Maps. an Example: the Froeschlé Map....Pages 403-407 A Many-Particle Adiabatic Invariant....Pages 408-412 The Non-Integrability of Perturbed Hamiltonians of N Degrees of Freedom and the Continuation of Periodic Orbits....Pages 413-417 Structure of the 5-Dimensional Manifold of Constant Energy in the J 2 -Problem....Pages 418-421 The Bifurcations of the Langmuir Orbit in the Two-Electron Atom....Pages 422-425 Domains of Convergence of Kam Type Iterations for Eigenvalue Problems....Pages 426-429 Nonlinear Dynamics in an Extended Neighbourhood of the Translunar Equilibrium Point....Pages 430-434 Analysis of Quantum Eigenstates in a 3-Mode System....Pages 435-439 Detection of Stable Manifolds in High Dimensional Phase Space....Pages 440-443 Ordered Motion in Hamiltonian Systems with Many Degrees of Freedom....Pages 444-448 Some Relations from Hamiltonian Mechanics and their Applications to Plasma Physics....Pages 449-452 Effective Stability in Reversible Systems....Pages 453-457 New Criterion of Nonintegrability for an N -Degrees-of-Freedom Hamiltonian System....Pages 458-470 The Invariant Measure for the Satellite Ground Station View Period Problem....Pages 471-474 The two Rigid Bodies Problem. Reduction and Relative Equilibria....Pages 475-479 Front Matter....Pages 283-283 Transition Orbits and Transition Times Along Chains of Hyperbolic Tori....Pages 480-484 Persistence of Asteroids After a Close Encounter....Pages 485-488 Structure of the Sets of Invariant Tori and Problems of Stability in Reversible Systems....Pages 489-493 On the Break-Up of Invariant Tori with three Frequencies....Pages 494-498 The Arnol’d Web in Atomic Physics....Pages 499-503 Psi-Series, Singularities of Solutions and Integrability of Polynomial Systems....Pages 504-508 Galoisian Obstructions to Integrability of Hamiltonian Systems: Statements and Examples....Pages 509-513 Bounds on Diffusion in Phase Space: Connection Between Nekhoroshev and Kam Theorems and Superexponential Stability of Invariant Tori....Pages 514-517 Bifurcation at Complex Instability....Pages 518-522 Global Dynamics of a Galactic Potential via Frequency Map Analysis....Pages 523-527 A Particular Class of Integrable Connection....Pages 528-532 Poincaré-Melnikov-Arnold Method for Twist Maps....Pages 533-537 Transport in a Class of N -D.O.F. Systems....Pages 538-543 Mel’nikov Vector and Singularity Analysis of Periodically Perturbed 2 D.O.F Hamiltonian Systems....Pages 544-548 A Study of Reduced Action-Functional for the Newtonian 3-Body Problem....Pages 549-553 Satellite Dynamics About Asteroids: Computing Poincaré Maps for the General Case....Pages 554-557 Mel’nikov Analysis of a Hamiltonian Perturbation of the Nonlinear Schrödinger Equation....Pages 558-562 First Order Adiabatic Approximation for a Class of Classical Slow-Fast Systems with Ergodic Fast Dynamics....Pages 563-567 The Lack-of-Parameters Problem in the Kam Theory Revisited....Pages 568-572 Homoclinic Invariant for the Coupled Standard Map....Pages 573-577 Front Matter....Pages 283-283 Destruction of Adiabatic Invariants on Resonances: Example from the Rigid Body Dynamics....Pages 578-582 Numerical Study of the Phase Space of a four Dimensional Symplectic Map....Pages 583-587 Caustics, Cusps and Chaos in Hamiltonian Systems....Pages 588-591 A Homoclinic Orbit for the Double Pendulum....Pages 592-594 Some Bifurcations Related to Homoclinic Tangencies for 1-Parameter Families of Symplectic Diffeomorphisms....Pages 595-599 Resonant Normal forms for four Dimensional Symplectic Mappings and Applications to Nonlinear Beam Dynamics....Pages 600-604 Approximation of Invariant Surfaces by Periodic Orbits in High-Dimensional Maps....Pages 605-609 Exponential Asymptotics and Approximation of Stable and Unstable Manifolds in Singularly Perturbed Nonlinear Systems....Pages 610-613 Singularity Analysis Towards Nonintegrability of Nonhomogeneous Nonlinear Lattices....Pages 614-617 Adiabatic Chaos of Streamlines in a Family of 3 D Confined Stokes Flows....Pages 618-624 Toda Lattices on Finite Symmetric Space Graphs....Pages 625-627 Effective Stability Around Periodic Orbits of the Spatial RTBP....Pages 628-632 On the Computation of Periodic Orbits and Invariant Surfaces of 4 D -Symplectic Mappings....Pages 633-637 On the Computation of all the Equilibrium Points in Hamiltonian Systems with Three Degrees of Freedom....Pages 638-641 A Method for Computing Families of Periodic Orbits Based on Unconstrained Optimization....Pages 642-645 Phase Space Geometry of Reactive Scattering....Pages 646-648 Numerical Construction of the Poincaré Map, with Application to Accelerators....Pages 649-653 Back Matter....Pages 654-658