Title Page
Copyright Page
Foreword
Preface
Table of Contents
Introduction
1. Numerical Method for the Newton Equation of Motion
(1) Calculation of the Harmonic oscillator’s elliptic orbit
(2) The elliptic orbit for the nonlinear oscillator
(3) The oval orbit of the Huygens oscillator
(4) The dense orbit of the geodesic for the ellipsoidal surface
(5) The close orbit of the geodesic for the ellipsoidal surface
(6) The close orbit of the Keplerian 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
(1) Applications of symplectic algorithms to large time scale system
(2) Applications of symplectic algorithms to qualitative analysis
(3) Applications of symplectic algorithms to quantitative computations
(4) Applications of symplectic algorithms to quantum systems
(5) Applications to computation of classical trajectories
(6) Applications to computation of classical trajectories of diatomic system [Dea94,DLea96]
(7) Applications to atmospheric and geophysical science
Bibliography
Chapter 1. Preliminaries of Differentiable Manifolds
1.1 Differentiable Manifolds
1.1.1 Differentiable Manifolds and Differentiable Mapping
1.1.2 Tangent Space and Differentials
1. Definition and properties of differentials of mappings
2. Geometrical meaning of differential of mappings
1.1.3 Submanifolds
1. Inverse function theorem
2. Immersion
3. Regular submanifolds
4. Embedded submanifolds
1.1.4 Submersion and Transversal
1.2 Tangent Bundle
1.2.1 Tangent Bundle and Orientation
1. Tangent Bundle
2. Orientation
1.2.2 Vector Field and Flow
1.3 Exterior Product
1.3.1 Exterior Form
1. 1- Form
2. 2-Forms
3. k-Forms
1.3.2 Exterior Algebra
1. The exterior product of two 1-forms
2. Exterior monomials
3. Exterior product of forms
1.4 Foundation of Differential Form
1.4.1 Differential Form
1.4.2 The Behavior of Differential Forms under Maps
1.4.3 Exterior Differential
1.4.4 Poincare Lemma and Its Inverse Lemma
1.4.5 Differential Form in R3
1.4.6 Hodge Duality and Star Operators
1.4.7 Codifferential Operator δ
1.4.8 Laplace–Beltrami Operator
1.5 Integration on a Manifold
1.5.1 Geometrical Preliminary
1.5.2 Integration and Stokes Theorem
1.5.3 Some Classical Theories on Vector Analysis
1.6 Cohomology and Homology
1.7 Lie Derivative
1.7.1 Vector Fields as Differential Operator
1.7.2 Flows of Vector Fields
1.7.3 Lie Derivative and Contraction
Bibliography
Chapter 2. Symplectic Algebra and Geometry Preliminaries
2.1 Symplectic Algebra and Orthogonal Algebra
2.1.1 Bilinear Form
1. Bilinear form
2. Quadratic forms induced by bilinear form
2.1.2 Sesquilinear Form
1. Sesquilinear form
2. Hermitian forms induced by sesquilinear forms
2.1.3 Scalar Product, Hermitian Product
2.1.4 Invariant Groups for Scalar Products
2.1.5 Real Representation of Complex Vector Space
2.1.6 Complexification of Real Vector Space and Real Linear Transformation
2.1.7 Lie Algebra for GL(n,F)
1. Lie algebra
2. Exponential matrix transform
3. Lie algebra of conformally invariant groups
2.2 Canonical Reductions of Bilinear Forms
2.2.1 Congruent Reductions
2.2.2 Congruence Canonical Forms of Conformally Symmet-ric and Hermitian Matrices
1. Alternative canonical forms
2. Invariants under congruences
2.2.3 Similar Reduction to Canonical Forms under Orthogo-nal Transformation
1. Elementary divisors in complex space
2. Elementary divisor in real space
2.3 Symplectic Space
2.3.1 Symplectic Space and Its Subspace
1. Comparison between symplectic and Euclidian space[Tre75,LM87,FQ91a,Wei77]
2. Special classes of subspaces[HW63,Tre75,LM87,Wei77]
3. Matrix representation of subspaces in R2n
2.3.2 Symplectic Group
2.3.3 Lagrangian Subspaces
2.3.4 Special Types of Sp(2n)
1. Several special types
2. Some theorems about Sp(2n)
2.3.5 Generators of Sp(2n)
2.3.6 Eigenvalues of Symplectic and Infinitesimal Matrices
2.3.7 Generating Functions for Lagrangian Subspaces
2.3.8 Generalized Lagrangian Subspaces
Bibliography
Chapter 3. Hamiltonian Mechanics and Symplectic Geometry
3.1 Symplectic Manifold
3.1.1 Symplectic Structure on Manifolds
3.1.2 Standard Symplectic Structure on Cotangent Bundles
3.1.3 Hamiltonian Vector Fields
3.1.4 Darboux Theorem
3.2 Hamiltonian Mechanics on R2n
3.2.1 Phase Space on R2n and Canonical Systems
1. 1-form and 2-form in R2n
2. Hamiltonian vector fields on R2n
3. Canonical systems
4. Integral invariants[Arn89]
3.2.2 Canonical Transformation
3.2.3 Poisson Bracket
1. Poisson bracket
2. Lie algebras of Hamiltonian vector fields and functions
3.2.4 Generating Functions
3.2.5 Hamilton–Jacobi Equations
Bibliography
Chapter 4. Symplectic Difference Schemes for Hamiltonian Systems
4.1 Background
4.1.1 Element and Notation for Hamiltonian Mechanics
4.1.2 Geometrical Meaning of Preserving Symplectic Struc-ture ω
4.1.3 Some Properties of a Symplectic Matrix
4.2 Symplectic Schemes for Linear Hamiltonian Systems
4.2.1 Some Symplectic Schemes for Linear Hamiltonian Sys-tems
4.2.2 Symplectic Schemes Based on Pade ´Approximation
4.2.3 Generalized Cayley Transformation and Its Application
4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian Sys-tem
4.4 Explicit Symplectic Scheme for Hamiltonian System
4.4.1 Systems with Nilpotent of Degree 2
4.4.2 Symplectically Separable Hamiltonian Systems
4.4.3 Separability of All Polynomials in R2n
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 Schemes
5.8 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
7.1.1 Definition and Properties of Symplectic R–K Method
7.1.2 Symplectic Conditions for R–K Method
7.1.3 Diagonally Implicit Symplectic R–K Method
7.1.4 Rooted Tree Theory
1. High order derivatives and rooted tree theory
2. Labeled graph
3. Relationship between rooted tree and elementary differential
4. Order conditions for multi-stage R–K method
7.1.5 Simplified Conditions for Symplectic R–K Method
7.2 Symplectic P–R–K Method
7.2.1 P–R–K Method
7.2.2 Symplified Order Conditions of Explicit Symplectic R–K Method
7.3 Symplectic R–K–N Method
7.3.1 Order Conditions for Symplectic R–K–N Method
7.3.2 The 3-Stage and 4-th order Symplectic R–K–N Method .
7.3.3 Symplified Order Conditions for Symplectic R–K–N Method
7.4 Formal Energy for Symplectic R–K Method
7.4.1 Modified Equation
7.4.2 Formal Energy for Symplectic R–K Method
7.5 Definition of a(t) and b(t)
7.5.1 Centered Euler Scheme
7.5.2 Gauss–Legendre Method
7.5.3 Diagonal Implicit R–K Method
7.6 Multistep Symplectic Method
7.6.1 Linear Multistep Method
7.6.2 Symplectic LMM for Linear Hamiltonian Systems
7.6.3 Rational Approximations to Exp and Log Function
1. Leap-frog scheme
2. Exponential function
3. Logarithmic function
4. Obreschkoff formula
5. Nonexistence of SLMM for Nonlinear Hamiltonian Systems (Tang Theorem)
Bibliography
Chapter 8. Composition Scheme
8.1 Construction of Fourth Order with 3-Stage Scheme
8.1.1 For Single Equation
8.1.2 For System of Equations
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
9.3.1 Approximations of Phase Flows and Numerical Method
9.3.2 Typical Algorithm and Step Transition Map
9.4 Related B-Series Works
9.4.1 The Composition Laws
9.4.2 Substitution Law
9.4.3 The Logarithmic Map
Bibliography
Chapter 10. Volume-Preserving Methods for Source-Free Systems
10.1 Liouville’s Theorem
10.2 Volume-Preserving Schemes
10.2.1 Conditions for Centered Euler Method to be Volume Preserving
10.2.2 Separable Systems and Volume-Preserving Explicit Meth-ods
10.3 Source-Free System
10.4 Obstruction to Analytic Methods
10.5 Decompositions of Source-Free Vector Fields
10.6 Construction of Volume-Preserving Schemes
10.7 Some Special Discussions for Separable Source-Free Systems
10.8 Construction of Volume-Preserving Scheme via Generating Function
10.8.1 Fundamental Theorem
10.8.2 Construction of Volume-Preserving Schemes
10.9 Some Volume-Preserving Algorithms
10.9.1 Volume-Preserving R–K Methods
10.9.2 Volume-Preserving 2-Stage P–R–K Methods
10.9.3 Some Generalizations
10.9.4 Some Explanations
Bibliography
Chapter 11. Contact Algorithms for Contact Dynamical Systems
11.1 Contact Structure
11.1.1 Basic Concepts of Contact Geometry
11.1.2 Contact Structure
11.2 Contactization and Symplectization
11.3 Contact Generating Functions for Contact Maps
11.4 Contact Algorithms for Contact Systems
11.4.1 Q Contact Algorithm
11.4.2 P Contact Algorithm
11.4.3 C Contact Algorithm
11.5 Hamilton–Jacobi Equations for Contact Systems
Bibliography
Chapter 12. Poisson Bracket and Lie–Poisson Schemes
12.1 Poisson Bracket and Lie–Poisson Systems
12.1.1 Poisson Bracket
1. Bilinearity
2. Skew-Symmetry
3. Jacobi Identity
4. Leibniz Rule
12.1.2 Lie–Poisson Systems
12.1.3 Introduction of the Generalized Rigid Body Motion
12.2 Constructing Difference Schemes for Linear Poisson Systems
12.2.1 Constructing Difference Schemes for Linear Poisson Systems
12.2.2 Construction of Difference Schemes for General Poisson Manifold
12.2.3 Answers of Some Questions
1. Euler explicit scheme[LQ95a]
2. Midpoint scheme and Euler scheme
12.3 Generating Function and Lie–Poisson Scheme
12.3.1 Lie–Poisson–Hamilton–Jacobi (LPHJ) Equation and Gen-erating Function
12.3.2 Construction of Lie–Poisson Schemes via Generating Function
12.4 Construction of Structure Preserving Schemes for Rigid Body
12.4.1 Rigid Body in Euclidean Space
12.4.2 Energy-Preserving and Angular Momentum-Preserving Schemes for Rigid Body
12.4.3 Orbit-Preserving and Angular-Momentum-Preserving Ex-plicit Scheme
12.4.4 Lie–Poisson Schemes for Free Rigid Body
12.4.5 Lie–Poisson Scheme on Heavy Top
12.4.6 Other Lie–Poisson Algorithm
1. Constrained Hamiltonian algorithm
2. Veselov–Moser algorithm
3. Reduction method
12.5 Relation Among Some Special Group and Its Lie Algebra
12.5.1 Relation Among SO(3), so(3) and SH1, SU(2)
1. Transformation between SO(3) and SH1
2. Relation between so(3) and SO(3)
3. Transformation between SO(3) and SH1
12.5.2 Representations of Some Functions in SO(3)
Bibliography
Chapter 13. KAM Theorem of Symplectic Algorithms
13.1 Brief Introduction to Stability of Geometric Numerical Algorithms
13.2 Mapping Version of the KAM Theorem
13.2.1 Formulation of the Theorem
13.2.2 Outline of the Proof of the Theorems
13.2.3 Application to Small Twist Mappings
13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems
13.3.1 Symplectic Algorithms as Small Twist Mappings
13.3.2 Numerical Version of KAM Theorem
13.4 Resonant and Diophantine Step Sizes
13.4.1 Step Size Resonance
13.4.2 Diophantine Step Sizes
13.4.3 Invariant Tori and Further Remarks
Bibliography
Chapter 14. Lee-Variational Integrator
14.1 Total Variation in Lagrangian Formalism
14.1.1 Variational Principle in Lagrangian Mechanics
14.1.2 Total Variation for Lagrangian Mechanics
14.1.3 Discrete Mechanics and Variational Integrators
14.1.4 Concluding Remarks
14.2 Total Variation in Hamiltonian Formalism
14.2.1 Variational Principle in Hamiltonian Mechanics
14.2.2 Total Variation in Hamiltonian Mechanics
14.2.3 Symplectic-Energy Integrators
14.2.4 High Order Symplectic-Energy Integrator
14.2.5 An Example and an Optimization Method
14.2.6 Concluding Remarks
14.3 Discrete Mechanics Based on Finite Element Methods
14.3.1 Discrete Mechanics Based on Linear Finite Element
14.3.2 Discrete Mechanics with Lagrangian of High Order
14.3.3 Time Steps as Variables
14.3.4 Conclusions
Bibliography
Chapter 15. Structure Preserving Schemes for Birkhoff Systems
15.1 Introduction
15.2 Birkhoffian Systems
15.3 Generating Functions for K(z, t)-Symplectic Mappings
15.4 Symplectic Difference Schemes for Birkhoffian Systems
15.5 Example
15.6 Numerical Experiments
Bibliography
Chapter 16. Multisymplectic and Variational Integrators
16.1 Introduction
16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Systems
1. Multisymplectic geometry
2. Multisymplectic Hamiltonian systems
16.3 Multisymplectic Integrators and Composition Methods
16.4 Variational Integrators
16.5 Some Generalizations
1. Multisymplectic Fourier pseudospectral methods
2. Nonconservative multisymplectic Hamiltonian systems
3. Construction of multisymplectic integrators for modified equations
4. Multisymplectic Birkhoffian systems
5. Differential complex, methods and multisymplectic structure
Bibliography
Symbol
Index