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Geography of Order and Chaos in Mechanics : Investigations of Quasi-Integrable Systems with Analytical, Numerical, and Graphical Tools / by Bruno Cordani
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Progress in Mathematical Physics ; 64 | SpringerLink BücherPublisher: New York, NY : Birkhäuser, 2013Description: Online-Ressource (XVIII, 332 p. 75 illus., 27 illus. in color, digital)ISBN:- 9780817683702
- 1283909928
- 9781283909921
- 530.15
- QA401-425 QC19.2-20.85
- QA401-425
- QC19.2-20.85
Contents:
Summary: Preface -- List of Figures -- 1 Introductory Survey -- 2 Analytical Mechanics and Integrable Systems -- 3 Perturbation Theory -- 4 Numerical Tools I: ODE Integration -- 5 Numerical Tools II: Detecting Order, Chaos, and Resonances -- 6 The Kepler Problem -- 7 The KEPLER Program -- 8 Some Perturbed Keplerian Systems -- 9 The Multi-Body Gravitational Problem -- Bibliography -- Index.Summary: This original monograph aims to explore the dynamics in the particular but very important and significant case of quasi-integrable Hamiltonian systems, or integrable systems slightly perturbed by other forces. With both analytic and numerical methods, the book studies several of these systems—including for example the hydrogen atom or the solar system, with the associated Arnold web—through modern tools such as the frequency-modified fourier transform, wavelets, and the frequency-modulation indicator. Meanwhile, it draws heavily on the more standard KAM and Nekhoroshev theorems. Geography of Order and Chaos in Mechanics contains many figures that illuminate its concepts in novel ways, but perhaps its most useful feature is its inclusion of software to reproduce the various numerical experiments. The graphical user interfaces of five supplied MATLAB programs allows readers without any knowledge of computer programming to visualize and experiment with the distribution of order, chaos and resonances in various Hamiltonian systems. This monograph will be a valuable resource for professional researchers and certain advanced undergraduate students in mathematics and physics, but mostly will be an exceptional reference for Ph.D. students with an interest in perturbation theory.PPN: PPN: 1651869529Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Geography of Order and Chaos in Mechanics; Preface; Contents; List of Figures; CHAPTER 1 Introductory Survey; 1.1 Configuration Space and Lagrangian Dynamics; 1.2 Symplectic Manifolds; 1.3 Phase Space and Hamiltonian Dynamics; 1.4 The Liouville and Arnold Theorems; 1.5 Quasi-Integrable Hamiltonian Systems and KAM Theorem; 1.6 Geography of the Phase Space; 1.7 Numerical Tools; 1.8 The Perturbed Kepler Problem; 1.9 The Multi-Body Gravitational Problem; CHAPTER 2 Analytical Mechanics and Integrable Systems; 2.1 Differential Geometry; 2.1.1 Differentiable Manifolds; Tangent and Cotangent Spaces
Push-forward and Pull-back2.1.2 Tensors and Forms; Forms and Exterior Derivative; Lie Derivative; 2.1.3 Riemannian, Symplectic, and Poisson Manifolds; Riemannian Manifolds; Symplectic Manifolds; Poisson Manifolds; 2.2 Lie Groups and Lie Algebras; 2.2.1 Definition and Properties; 2.2.2 Adjoint and Coadjoint Representation; 2.2.3 Action of a Lie Group on a Manifold; 2.2.4 Classification of Lie Groups and Lie Algebras; 2.3 Lagrangian and Hamiltonian Mechanics; 2.3.1 Lagrange Equations; 2.3.2 Hamilton Principle; 2.3.3 Noether Theorem; 2.3.4 From Lagrange to Hamilton
2.3.5 Canonical Transformations2.3.6 Hamilton-Jacobi Equation; 2.3.7 Symmetries and Reduction; 2.3.8 Liouville Theorem; 2.3.9 Arnold Theorem; 2.3.10 Action-Angle Variables: Examples; 2.A Appendix: The Problem of two Fixed Centers; CHAPTER 3 Perturbation Theory; 3.1 Formal Expansions; 3.1.1 Lie Series and Formal Canonical Transformations; 3.1.2 Homological Equation and Its Formal Solution; 3.2 Perpetual Stability and KAM Theorem; 3.2.1 Cauchy Inequality; 3.2.2 Convergence of Lie Series; 3.2.3 Homological Equation and Its Solution; 3.2.4 KAM Theorem (According to Kolmogorov)
3.2.5 KAM Theorem (According to Arnold)3.2.6 Isoenergetic KAM Theorem; 3.3 Exponentially Long Stability and Nekhoroshev Theorem; 3.4 Geography of the Phase Space; 3.5 Elliptic Equilibrium Points; 3.A Appendix: Results from Diophantine Theory; 3.B Appendix: Homoclinic Tangle and Chaos; CHAPTER 4 Numerical Tools I: ODE Integration; 4.1 Cauchy Problem; 4.2 Euler Method; 4.3 Runge-Kutta Methods; 4.4 Gragg-Bulirsch-Stoer Method; 4.5 Adams-Bashforth-Moulton Methods; 4.6 Geometric Methods; 4.7 What Methods Are in the MATLAB Programs?; ode113; ode45; IRK-Gauss; Dop853; Odex; Tom; RungeKutta4
RungeKutta5gni_irk2; gni_lmm2; gni_comp; CHAPTER 5 Numerical Tools II: Detecting Order, Chaos, and Resonances; 5.1 Poincaré Section; 5.2 Variational Equation Methods; 5.2.1 The Largest Lyapunov Exponents; 5.2.2 The Fast Lyapunov Indicator; 5.2.3 Other Methods; 5.3 Fourier Transform Methods; 5.3.1 Fast Fourier Transform (FFT); 5.3.2 Frequency Modified Fourier Transform (FMFT); 5.3.3 Wavelets and Time-Frequency Analysis; 5.3.4 Frequency Modulation Indicator (FMI); 5.4 Some Examples; 5.4.1 Symplectic Maps and the POINCARE Program; 5.4.2 A Test Hamiltonian and the HAMILTON Program
5.4.3 The Lagrange Points L4 and L5 and the LAGRANGE Program
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