Format:
Online-Ressource (XXIV, 676p. 62 illus, digital)
ISBN:
9783642017773
Series Statement:
SpringerLink
Content:
Preliminaries of Differentiable Manifolds -- Symplectic Algebra and Geometry Preliminaries -- Hamiltonian Mechanics and Symplectic Geometry -- Symplectic Difference Schemes for Hamiltonian Systems -- The Generating Function Method -- The Calculus of Generating Functions and Formal Energy -- Symplectic Runge-Kutta Methods -- Composition Scheme -- Formal Power Series and B-Series -- Volume-Preserving Methods for Source-Free Systems -- Contact Algorithms for Contact Dynamical Systems -- Poisson Bracket and Lie-Poisson Schemes -- KAM Theorem of Symplectic Algorithms -- Lee-Variational Integrator -- Structure Preserving Schemes for Birkhoff Systems -- Multisymplectic and Variational Integrators.
Content:
"Symplectic Geometric Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications.
Note:
Includes bibliographical references and index
,
Title Page; Copyright Page; Foreword; Preface; Table of Contents; Introduction; 1. Numerical Method for the Newton Equation of Motion; 2. History of the Hamiltonian Mechanics; 3. The Importance of the Hamiltonian System; 4. Technical Approach-Symplectic Geometry Method; 5. The Symplectic Schemes; 6. The Volume-Preserving Scheme for Source-free System; 7. The Contact Schemes for Contact System; 8. Applications of the Symplectic Algorithms for Dynamics System; Bibliography; Chapter 1. Preliminaries of Differentiable Manifolds; 1.1 Differentiable Manifolds; 1.2 Tangent Bundle
,
1.3 Exterior Product1.4 Foundation of Differential Form; 1.5 Integration on a Manifold; 1.6 Cohomology and Homology; 1.7 Lie Derivative; Bibliography; Chapter 2. Symplectic Algebra and Geometry Preliminaries; 2.1 Symplectic Algebra and Orthogonal Algebra; 2.2 Canonical Reductions of Bilinear Forms; 2.3 Symplectic Space; Bibliography; Chapter 3. Hamiltonian Mechanics and Symplectic Geometry; 3.1 Symplectic Manifold; 3.2 Hamiltonian Mechanics on R2n; Bibliography; Chapter 4. Symplectic Difference Schemes for Hamiltonian Systems; 4.1 Background
,
4.2 Symplectic Schemes for Linear Hamiltonian Systems4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian Sys-tem; 4.4 Explicit Symplectic Scheme for Hamiltonian System; 4.5 Energy-conservative Schemes by Hamiltonian Difference; Bibliography; Chapter 5. The Generating Function Method; 5.1 Linear Fractional Transformation; 5.2 Symplectic, Gradient Mapping and Generating Function; 5.3 Generating Functions for the Phase Flow; 5.4 Construction of Canonical Difference Schemes; 5.5 Further Remarks on Generating Function; 5.6 Conservation Laws
,
5.7 Convergence of Symplectic Difference Schemes5.8 Symplectic Schemes for Nonautonomous System; Bibliography; Chapter 6. The Calculus of Generating Functions and Formal Energy; 6.1 Darboux Transformation; 6.2 Normalization of Darboux Transformation; 6.3 Transform Properties of Generator Maps and Generating Functions; 6.4 Invariance of Generating Functions and Commutativity of Generator Maps; 6.5 Formal Energy for Hamiltonian Algorithm; 6.6 Ge-Marsden Theorem; Bibliography; Chapter 7. Symplectic Runge-Kutta Methods; 7.1 Multistage Symplectic Runge-Kutta Method; 7.2 Symplectic P-R-K Method
,
7.3 Symplectic R-K-N Method7.4 Formal Energy for Symplectic R-K Method; 7.5 Definition of a(t) and b(t); 7.6 Multistep Symplectic Method; Bibliography; Chapter 8. Composition Scheme; 8.1 Construction of Fourth Order with 3-Stage Scheme; 8.2 Adjoint Method and Self-Adjoint Method; 8.3 Construction of Higher Order Schemes; 8.4 Stability Analysis for Composition Scheme; 8.5 Application of Composition Schemes to PDE; 8.6 H-Stability of Hamiltonian System; Bibliography; Chapter 9. Formal Power Series and B-Series; 9.1 Notation; 9.2 Near-0 and Near-1 Formal Power Series
,
9.3 Algorithmic Approximations to Phase Flows
Additional Edition:
ISBN 9783642017766
Additional Edition:
Buchausg. u.d.T. Feng, Kang, 1920 - 1993 Symplectic geometric algorithms for Hamiltonian systems Berlin : Springer, 2010 ISBN 9787534135958
Additional Edition:
ISBN 9783642017766
Language:
English
Subjects:
Mathematics
Keywords:
Hamiltonsches System
;
Symplektische Geometrie
;
Hamiltonsches System
;
Symplektische Geometrie
DOI:
10.1007/978-3-642-01777-3
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Qin, Mengzhao
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