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Hamiltonian Systems with Three or More Degrees of Freedom : NATO Science Series C - Charles Simo

Hamiltonian Systems with Three or More Degrees of Freedom

NATO Science Series C

By: Charles Simo (Editor)


Published: 30th June 1999
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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 Schr‚ÄĚdinger 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.

Inflection Points, Extatic Points and Curve Shorteningp. 3
Topologically Necessary Singularities on Moving Wavefronts and Causticsp. 11
Heteroclinic Chains of Skew Product Hamiltonian Systemsp. 13
Order and Chaos in 3-D Systemsp. 26
Splitting of Separatrices in Hamiltonian Systems and Symplectic Mapsp. 39
On the Discrete One-Dimensional Quasi-Periodic Schrodinger Equation and Other Smooth Quasi-Periodic Skew Productsp. 55
Lindstedt Series and Kolmogorov Theoremp. 62
A Classical Self-Contained Proof of Kolmogorov's Theorem on Invariant Torip. 72
Dynamical Stability in Lagrangian Systemsp. 90
The Origin of Chaotic Behaviour in the Kirkwood Gapsp. 115
Examples of Compact Hypersurfaces in R[superscript 2P], 2P [actual symbol not reproducible] 6, with no Periodic Orbitsp. 126
Hamiltonian Systems with Three Degrees of Freedom and Hydrodynamicsp. 127
Introduction to Frequency Map Analysisp. 134
Lindstedt Series for Lower Dimensional Torip. 151
Arnold Diffusion; A Compendium of Remarks and Questionsp. 168
Old and New Applications of Kam Theoryp. 184
On Adiabatic Invariance in Two-Frequency Systemsp. 193
The Method of Rational Approximations: Theory and Applicationsp. 213
Dynamical Systems Methods for Space Missions on a Vicinity of Collinear Libration Pointsp. 223
A Mechanism of Ergodicity in Standard-Like Mapsp. 242
Continuous Averaging in Hamiltonian Systemsp. 244
Phase Space Geometry and Dynamics Associated with the 1 : 2 : 2 Resonancep. 254
From Singular Point Analysis to Rigorous Results on Integrability: A Dream of S. Kowalevskayap. 270
Time Singularities for Polynomial Hamiltonians with Analytic Time Dependencep. 285
Numerical Study of Turbulence in N-Body Hamiltonian Systems with Long Range Force I. Relaxationp. 290
Phase Space Structures in 3 and 4 Degrees of Freedom: Application to Chemical Reactionsp. 295
Modulated Diffusion for Symplectic Mapsp. 300
On the Jeans-Landau-Teller Approximation for Adiabatic Invariantsp. 305
Periodic Orbits and Quantum Mechanics of Molecular Hamiltonian Systemsp. 310
Chaos in Atom-Surface Collisionsp. 314
On the Non-Integrability of the Mixmaster Universe Modelp. 318
Branching of Solutions as Obstruction to the Existence of a Meromorphic Integral in Many-Dimensional Systemsp. 324
The Energy Surfaces of the Kovalevskaya-Topp. 330
Filling Rates for Linear Flow on the Torus: Recent Progress and Applicationsp. 335
Invariant Spectra of Orbits in Multidimensional Symplectic Mapsp. 340
On the Convergence of Formal Series Containing Small Divisorsp. 345
Systems with an Invariant Measure on Lie Groupsp. 350
Stochasticity of the 2/1 Asteroidal Resonance: A Symplectic Mapping Approachp. 357
On Perturbed Oscillators in 1 - 1 - 1 Resonance: Critical Inclination in the 3D Henon-Heiles Potentialp. 362
Splitting of Separatrices for (Fast) Quasiperiodic Forcingp. 367
A Possible Mechanism for the Kam Tori Breakdownp. 372
On the Dynamics Near Resonant Equilibriap. 377
Exponentially Small Estimates for Kam Theorem Near an Elliptic Equilibrium Pointp. 386
Fast Diffusion and Universality Near Intersecting Resonancesp. 391
Quasi-Periodic Motion of a Rigid Body under Weak Forcesp. 398
Center and Center-(Un)Stable Manifolds of Elliptic-Hyperbolic Fixed Points of 4D-Symplectic Maps. An Example: The Froeschle Mapp. 403
A Many-Particle Adiabatic Invariantp. 408
The Non-Integrability of Perturbed Hamiltonians of N Degrees of Freedom and the Continuation of Periodic Orbitsp. 413
Structure of the 5-Dimensional Manifold of Constant Energy in the J[subscript 2]-Problemp. 418
The Bifurcations of the Langmuir Orbit in the Two-Electron Atomp. 422
Domains of Convergence of Kam Type Iterations for Eigenvalue Problemsp. 426
Nonlinear Dynamics in an Extended Neighbourhood of the Translunar Equilibrium Pointp. 430
Analysis of Quantum Eigenstates in a 3-Mode Systemp. 435
Detection of Stable Manifolds in High Dimensional Phase Spacep. 440
Ordered Motion in Hamiltonian Systems with Many Degrees of Freedomp. 444
Some Relations from Hamiltonian Mechanics and their Applications to Plasma Physicsp. 449
Effective Stability in Reversible Systemsp. 453
New Criterion of Nonintegrability for an N-Degrees-of-Freedom Hamiltonian Systemp. 458
The Invariant Measure for the Satellite Ground Station View Period Problemp. 471
The Two Rigid Bodies Problem. Reduction and Relative Equilibriap. 475
Transition Orbits and Transition Times Along Chains of Hyperbolic Torip. 480
Persistence of Asteroids After a Close Encounterp. 485
Structure of the Sets of Invariant Tori and Problems of Stability in Reversible Systemsp. 489
On the Break-Up of Invariant Tori with Three Frequenciesp. 494
The Arnol'd Web in Atomic Physicsp. 499
PSI-Series, Singularities of Solutions and Integrability of Polynomial Systemsp. 504
Galoisian Obstructions to Integrability of Hamiltonian Systems: Statements and Examplesp. 509
Bounds on Diffusion in Phase Space: Connection between Nekhoroshev and Kam Theorems and Superexponential Stability of Invariant Torip. 514
Bifurcation at Complex Instabilityp. 518
Global Dynamics of a Galactic Potential Via Frequency Map Analysisp. 523
A Particular Class of Integrable Connectionp. 528
Poincare-Melnikov-Arnold Method for Twist Mapsp. 533
Transport in a Class of N-D.O.F. Systemsp. 538
Mel'nikov Vector and Singularity Analysis of Periodically Perturbed 2 D.O.F Hamiltonian Systemsp. 544
A Study of Reduced Action-Functional for the Newtonian 3-Body Problemp. 549
Satellite Dynamics About Asteroids: Computing Poincare Maps for the General Casep. 554
Mel-Nikov Analysis of a Hamiltonian Perturbation of the Nonlinear Schrodinger Equationp. 558
First Order Adiabatic Approximation for a Class of Classical Slow-Fast Systems with Ergodic Fast Dynamicsp. 563
The Lack-of-Parameters Problem in the Kam Theory Revisitedp. 568
Homoclinic Invariant for the Coupled Standard Mapp. 573
Destruction of Adiabatic Invariants on Resonances: Example from the Rigid Body Dynamicsp. 578
Numerical Study of the Phase Space of a Four Dimensional Symplectic Mapp. 583
Caustics, Cusps and Chaos in Hamiltonian Systemsp. 588
A Homoclinic Orbit for the Double Pendulump. 592
Some Bifurcations Related to Homoclinic Tangencies for 1-Parameter Families of Symplectic Diffeomorphismsp. 595
Resonant Normal Forms for Four Dimensional Symplectic Mappings and Applications to Nonlinear Beam Dynamicsp. 600
Approximation of Invariant Surfaces by Periodic Orbits in High-Dimensional Mapsp. 605
Exponential Asymptotics and Approximation of Stable and Unstable Manifolds in Singularly Perturbed Nonlinear Systemsp. 610
Singularity Analysis Towards Nonintegrability of Nonhomogeneous Nonlinear Latticesp. 614
Adiabatic Chaos of Streamlines in a Family of 3D Confined Stokes Flowsp. 618
Toda Lattices on Finite Symmetric Space Graphsp. 625
Effective Stability Around Periodic Orbits of the Spatial RTBPp. 628
On the Computation of Periodic Orbits and Invariant Surfaces of 4D-Symplectic Mappingsp. 633
On the Computation of All the Equilibrium Points in Hamiltonian Systems with Three Degrees of Freedomp. 638
A Method for Computing Families of Periodic Orbits Based on Unconstrained Optimizationp. 642
Phase Space Geometry of Reactive Scatteringp. 646
Numerical Construction of the Poincare Map, with Application to Acceleratorsp. 649
List of authorsp. 654
Subject indexp. 655
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792357100
ISBN-10: 0792357108
Series: NATO Science Series C
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 658
Published: 30th June 1999
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 3.81
Weight (kg): 2.5