Dynamical Systems - Vol.3: Mathematical Aspects of Classical and Celestial Mechanics

Die grundlegenden Fragestellungen und Methoden der klassischen Mechanik werden in diesem Band dargestellt. Die Encyclopaedia of Mathematical Sciences gibt eine sorgfältige, umfassende Darstellung wichtiger Gebiete der Mathematik samt Anwendungen. Sie wird bald allen Mathematikern und Anwendern, sei es in Forschung, Lehre oder Industrie, ein unverzichtbares Arbeitsmittel sein.

List of contents

1. Basic Principles of Classical Mechanics.- 1. Newtonian Mechanics.- 1.1. Space, Time, Motion.- 1.2. The Newton-Laplace Principle of Determinacy.- 1.3. The Principle of Relativity.- 1.4. Basic Dynamical Quantities. Conservation Laws.- 2. Lagrangian Mechanics.- 2.1. Preliminary Remarks.- 2.2. Variations and Extremals.- 2.3. Lagrange's Equations.- 2.4. Poincare's Equations.- 2.5. Constrained Motion.- 3. Hamiltonian Mechanics.- 3.1. Symplectic Structures and Hamilton's Equations.- 3.2. Generating Functions.- 3.3. Symplectic Structure of the Cotangent Bundle.- 3.4. The Problem of n Point Vortices.- 3.5. The Action Functional in Phase Space.- 3.6. Integral Invariants.- 3.7. Applications to the Dynamics of Ideal Fluids.- 3.8. Principle of Stationary Isoenergetic Action.- 4. Vakonomic Mechanics.- 4.1. Lagrange's Problem.- 4.2. Vakonomic Mechanics.- 4.3. The Principle of Determinacy.- 4.4. Hamilton's Equations in Redundant Coordinates.- 5. Hamiltonian Formalism with Constraints.- 5.1. Dirac's Problem.- 5.2. Duality.- 6. Realization of Constraints.- 6.1. Various Methods of Realizing Constraints.- 6.2. Holonomic Constraints.- 6.3. Anisotropic Friction.- 6.4. Adjoining Masses.- 6.5. Adjoining Masses and Anisotropic Friction.- 6.6. Small Masses.- 2. The ?-Body Problem.- 1. The Two-Body Problem.- 1.1. Orbits.- 1.2. Anomalies.- 1.3. Collisions and Regularization.- 1.4. Geometry of the Kepler Problem.- 2. Collisions and Regularization.- 2.1. Necessary Conditions for Stability.- 2.2. Simultaneous Collisions.- 2.3. Binary Collisions.- 2.4. Singularities of Solutions in the ?-Body Problem.- 3. Particular Solutions.- 3.1. Central Configurations.- 3.2. Homographic Solutions.- 3.3. The Amended Potential and Relative Equilibria.- 4. Final Motions in the Three-Body Problem.- 4.1. Classification of Final Motions According to Chazy.- 4.2. Symmetry of Past and Future.- 5. The Restricted Three-Body Problem.- 5.1. Equations of Motion. The Jacobi Integral.- 5.2. Relative Equilibria and the Hill Region.- 5.3. Hill's Problem.- 6. Ergodic Theorems in Celestial Mechanics.- 6.1. Stability in the Sense of Poisson.- 6.2. Probability of Capture.- 3. Symmetry Groups and Reduction (Lowering the Order).- 1. Symmetries and Linear First Integrals.- 1.1. E. Noether's Theorem.- 1.2. Symmetries in Nonholonomic Mechanics.- 1.3. Symmetries in Vakonomic Mechanics.- 1.4. Symmetries in Hamiltonian Mechanics.- 2. Reduction of Systems with Symmetry.- 2.1. Lowering the Order (the Lagrangian Aspect).- 2.2. Lowering the Order (the Hamiltonian Aspect).- 2.3. Examples: Free Motion of a Rigid Body and the Three-Body Problem.- 3. Relative Equilibria and Bifurcations of Invariant Manifolds.- 3.1. Relative Equilibria and the Amended Potential.- 3.2. Invariant Manifolds, Regions of Possible Motions, and Bifurcation Sets.- 3.3. The Bifurcation Set in the Planar Three-Body Problem.- 3.4. Bifurcation Sets and Invariant Manifolds in the Motion of a Heavy Rigid Body with a Fixed Point.- 4. Integrable Systems and Integration Methods.- 1. Brief Survey of Various Approaches to the Integrability of Hamiltonian Systems.- 1.1. Quadratures.- 1.2. Complete Integrability.- 1.3. Normal Forms.- 2. Completely Integrable Systems.- 2.1. Action-Angle Variables.- 2.2. Noncommutative Sets of First Integrals.- 2.3. Examples of Completely Integrable Systems.- 3. Some Methods of Integrating Hamiltonian Systems.- 3.1. Method of Separation of Variables.- 3.2. Method of L-A (Lax) Pairs.- 4. Nonholonomic Integrable Systems.- 4.1. Differential Equations with Invariant Measure.- 4.2. Some Solved Problems of Nonholonomic Mechanics.- 5. Perturbation Theory for Integrable Systems.- 1. Averaging of Perturbations.- 1.1. The Averaging Principle.- 1.2. Procedure for Eliminating Fast Variables in the Absence of Resonances.- 1.3. Procedure for Eliminating Fast Variables in the Presence of Resonances.- 1.4. Averaging in Single-Frequency Systems.- 1.5. Averaging in Systems with Constant Frequencies.- 1.6. Averaging in Nonresonant Domains.- 1.7. The Effect of a Single Resonance.- 1.8. Averaging in Two-Frequency Systems.- 1.9. Averaging in Multi-Frequency Systems.- 2. Averaging in Hamiltonian Systems.- 2.1. Application of the Averaging Principle.- 2.2. Procedures for Eliminating Fast Variables.- 3. The KAM Theory.- 3.1. Unperturbed Motion. Nondegeneracy Conditions.- 3.2. Invariant Tori of the Perturbed System.- 3.3. Systems with Two Degrees of Freedom.- 3.4. Diffusion of Slow Variables in Higher-Dimensional Systems, and its Exponential Estimate.- 3.5. Variants of the Theorem on Invariant Tori.- 3.6. A Variational Principle for Invariant Tori. Cantori.- 3.7. Applications of the KAM Theory.- 4. Adiabatic Invariants.- 4.1. Adiabatic Invariance of the Action Variable in Single-Frequency Systems.- 4.2. Adiabatic Invariants of Multi-Frequency Hamiltonian Systems.- 4.3. Procedure for Eliminating Fast Variables. Conservation Time of Adiabatic Invariants.- 4.4. Accuracy of the Conservation of Adiabatic Invariants.- 4.5. Perpetual Conservation of Adiabatic Invariants.- 6. Nonintegrable Systems.- 1. Near-Integrable Hamiltonian Systems.- 1.1. Poincare's Methods.- 1.2. Creation of Isolated Periodic Solutions is an Obstruction to Integrability.- 1.3. Applications of Poincare's Method.- 2. Splitting of Asymptotic Surfaces.- 2.1. Conditions for Splitting.- 2.2. Splitting of Asymptotic Surfaces is an Obstruction to Integrability.- 2.3. Applications.- 3. Quasi-Random Oscillations.- 3.1. The Poincare Map.- 3.2. Symbolic Dynamics.- 3.3. Nonexistence of Analytic First Integrals.- 4. Nonintegrability in the Neighborhood of an Equilibrium Position (Siegel's Method).- 5. Branching of Solutions and Nonexistence of Single-Valued First Integrals.- 5.1. Branching of Solutions is an Obstruction to Integrability.- 5.2. Monodromy Groups of Hamiltonian Systems with Single-Valued First Integrals.- 6. Topological and Geometrical Obstructions to Complete Integrability of Natural Systems with Two Degrees of Freedom.- 6.1. Topology of the Configuration Space of Integrable Systems.- 6.2. Geometrical Obstructions to Integrability.- 7. Theory of Small Oscillations.- 1. Linearization.- 2. Normal Forms of Linear Oscillations.- 2.1. Normal Form of Linear Natural Lagrangian Systems.- 2.2. The Rayleigh-Fischer-Courant Theorems on the Behavior of Characteristic Frequencies under an Increase in Rigidity and under Imposition of Constraints.- 2.3. Normal Forms of Quadratic Hamiltonians.- 3. Normal Forms of Hamiltonian Systems Near Equilibria.- 3.1. Reduction to Normal Form.- 3.2. Phase Portraits of Systems with Two Degrees of Freedom in the Neighborhood of an Equilibrium Position under Resonance.- 3.3. Stability of Equilibria in Systems with Two Degrees of Freedom under Resonance.- 4. Normal Forms of Hamiltonian Systems Near Closed Trajectories.- 4.1. Reduction to the Equilibrium of a System with Periodic Coefficients.- 4.2. Reduction of Systems with Periodic Coefficients to Normal Form.- 4.3. Phase Portraits of Systems with two Degrees of Freedom Near a Closed Trajectory under Resonance.- 5. Stability of Equilibria in Conservative Fields.- Comments on the Bibliography.- Recommended Reading.

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From the reviews: "... As an encyclopaedia article, this book does not seek to serve as a textbook, nor to replace the original articles whose results it describes. The book's goal is to provide an overview, pointing out highlights and unsolved problems, and putting individual results into a coherent context. It is full of historical nuggets, many of them surprising. ... The examples are especially helpful; if a particular topic seems difficult, a later example frequently tames it. The writing is refreshingly direct, never degenerating into a vocabulary lesson for its own sake. The book accomplishes the goals it has set for itself. While it is not an introduction to the field, it is an excellent overview. ..." American Mathematical Monthly, Nov. 1989 "This is a book to curl up with in front of a fire on a cold winter's evening. ..." SIAM Reviews, Sept. 1989

Product details

• Authors: Vladimir Igorevic Arnold, V. V. Kozlov, A. I. Neishtadt
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.01.1997
• EAN: 9783540612247
• ISBN: 978-3-540-61224-7
• No. of pages: 291
• Weight: 574 g
• Illustrations: w. 81 figs.
• Sets: Dynamical Systems; Dynamical Systems
• Series: Encyclopaedia of Mathematical Sciences; Encyclopaedia of Mathematical Sciences
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis