Kang Feng Mengzhao

Symplectic Geometric Algorithms for Hamiltonian Systems

Kang Feng Mengzhao Qin

Symplectic Geometric Algorithms for Hamiltonian Systems

With 62 Figures

ZHEJIANG PUBLISHING UNITED GROUP ZHEJIANG SCIENCE AND TECHNOLOGY PUBLISHING HOUSE Authors Kang Feng (1920-1993) Mengzhao Qin Institute of Computational Institute of Computational and Scientific/ Mathematics and Scientific/ Engineering Computing Engineering Computing Beijing 100190, China Beijing 100190, China Email: [email protected]

ISBN 978-7-5341-3595-8 Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou

ISBN 978-3-642-01776-6 ISBN 978-3-642-01777-3 (eBook) Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2009930026

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Springer is part of Springer Science+Business Media (www.springer.com) “... In the late 1980s Feng Kang pro- posed and developed so-called symplec- tic algorithms for solving equations in Hamiltonian form. Combining theoreti- cal analysis and computer experimenta- tion, showed that such methods, over times, are much superior to standard methods. At the time of his death, he was at work on extensions of this idea to other structures ...”

Peter Lax

Cited from SIAM News November 1993 Kang Feng giving a talk at an international conference

“ A basic idea behind the design of nu- merical schemes is that they can preserve the properties of the original problems as much as possible ... Different represen- tations for the same physical law can lead to different computational techniques in solving the same problem, which can pro- duce different numerical results ...”

Kang Feng (1920 – 1993)

Cited from a paper entitled “How to compute property Newton’s equation of motion” Prize certificate

Author’s photograph taken in Xi’an in 1989 Foreword

Kang Feng (1920–1993), Member of the Chinese Academy of Sciences, Professor and Honorary Director of the Computing Center of the Chinese Academy of Sciences, famous applied , founder and pioneer of computational mathematics and scientific computing in China. It has been 16 years since my brother Kang Feng passed away. His scientific achievements have been recognized more and more clearly over time, and his contri- butions to various fields have become increasingly outstanding. In the spring of 1997, Professor Shing-Tung Yau, a winner of the Fields Medal and a foreign member of the Chinese Academy of Sciences, mentioned in a presentation at , entitled “The development of mathematics in China in my view”, that “there are three main reasons for Chinese modern mathematics to go beyond or hand in hand with the West. Of course, I am not saying that there are no other works, but I mainly talk about the mathematics that is well known historically: Professor Shiingshen Chern’s work on characteristic class, Luogeng Hua’s work on the theory of functions of several complex variables, and Kang Feng’s work on finite elements.” This high evaluation of Kang Feng as a mathematician (not just a computational mathematician) sounds so refreshing that many people talked about it and strongly agreed with it. At the end of 1997, the Chinese National Natural Science Foundation presented Kang Feng et al. with the first class prize for his other work on a symplectic algorithm for Hamil- tonian systems, which is a further recognition of his scientific achievements (see the certificate on the previous page). As his brother, I am very pleased. Achieving a major scientific breakthrough is a rare event. It requires vision, ability and opportunity, all of which are indispensable. Kang Feng has achieved two major scientific breakthroughs in his life, both of which are very valuable and worthy of mention. Firstly, from 1964 to 1965, he proposed independently the finite element method and laid the foundation for the mathematical theory. Secondly, in 1984, he proposed a symplectic algorithm for Hamiltonian systems. At present, scientific inno- vation has become the focus of discussion. Kang Feng’s two scientific breakthroughs may be treated as case studies in scientific innovation. It is worth emphasizing that these breakthroughs were achieved in China by Chinese scientists. Careful study of these has yet to be carried out by experts. Here I just describe some of my personal feelings. It should be noted that these breakthroughs resulted not only from the profound mathematical knowledge of Kang Feng, but also from his expertise in classical physics and engineering technology that were closely related to the projects. Scientific break- throughs are often cross-disciplinary. In addition, there is often a long period of time before a breakthrough is made-not unlike a long time it takes for a baby to be born, which requires the accumulation of results in small steps. x Foreword

The opportunity for inventing the finite element method came from a national re- search project, a computational problem in the design of the dam. For such a concrete problem, Kang Feng found a basis for solving of the problem using his sharp insight. In his view, a discrete computing method for a mathematical and physical problem is usually carried out in four steps. Firstly, one needs to know and define the physical mechanism. Secondly, one writes the appropriate differential equa- tions accordingly. In the third step, design a discrete model. Finally, one develops the numerical algorithm. However, due to the complexity of the geometry and physical conditions, conventional methods cannot always be effective. Nonetheless, starting from the physical law of conservation or variational principle of the matter, we can directly relate to the appropriate discrete model. Combining the variational principle with the spline approximation leads to the finite element method, which has a wide range of adaptability and is particularly suited to deal with the complex geometry of the physical conditions of computational engineering problems. In 1965, Kang Feng published his paper entitled “Difference schemes based on the variational principle”, which solved the basic theoretical issues of the finite element method, such as conver- gence, error estimation, and stability. It laid the mathematical foundation for the finite element method. This paper is the main evidence for recognition by the international academic community of our independent development of the finite element method. After the Chinese Cultural Revolution, he continued his research in finite element and related areas. During this period, he made several great achievements. I remem- ber that he talked with me about other issues, such as Thom’s catastrophe theory, Prigogine’s theory of dissipative structures, solitons in water waves, the Radon trans- form, and so on. These problems are related to physics and engineering technology. Clearly he was exploring for new areas and seeking a breakthrough. In the 1970s, Arnold’s “Mathematical Method of Classical Mechanics” came out. It described the symplectic structure for Hamiltonian equations, which proved to be a great inspira- tion to him and led to a breakthrough. Through his long-term experience in mathe- matical computation, he fully realized that different mathematical expressions for the same physical law, which are physically equivalent, can perform different functions in scientific computing (his students later called this the “Feng’s major theorem”). In this way, for classical mechanics, Newton’s equations, Lagrangian equations and Hamiltonian equations will show a different pattern of calculations after discretiza- tion. Because the Hamiltonian formulation has a symplectic structure, he was keenly aware that, if the algorithm can maintain the geometric symmetry of symplecticity, it will be possible to avoid the flaw of artificial dissipation of this type of algorithm and design a high-fidelity algorithm. Thus, he opened up a broad way for the computa- tional method of the Hamiltonian system. He called this way the “Hamiltonian way”. This computational method has been used in the calculation of the orbit in celestial mechanics, in calculations for the particle path in accelerator, as well as in molecular dynamics. Later, the scope of its application was expanded. For example, it has also been widely used in studies of the atmosphere and earth sciences and elsewhere. It Foreword xi has been effectively applied in solving the GPS observation operator, indicating that Global Positioning System data can be dealt with in a timely manner. This algorithm is 400 times more efficient than the traditional method. In addition, a symplectic al- gorithm has been successfully used in the oil and gas exploration fields. Under the influence of Kang Feng, international research on symplectic algorithm has become popular and flourishing, nearly 300 papers have been published in this field to date. Kang Feng’s research work on the symplectic algorithm has been well-known and recognized internationally for its unique, innovative, systematic and widespread prop- erties, for its theoretical integrity and fruitful results. J. Lions, the former President of the International Mathematics Union, spoke at a workshop when celebrating his 60th birthday: “This is another major innovation made by Kang Feng, independent of the West, after the finite element method.” In 1993 one of the world’s leading , P.D. Lax, a member of the Ameri- can Academy of Sciences, wrote a memorial article dedicated to Kang Feng in SIAM News, stating that “In the late 1980s, Kang Feng proposed and developed so-called symplectic algorithms for solving evolution equations .... Such methods, over a long period, are much superior to standard methods.” E. J. Marsden, an internationlly well- known applied mathematician, visited the computing institute in the late 1980s and had a long conversation with Kang Feng. Soon after the death of Kang Feng, he pro- posed the multi-symplectic algorithm and extended the characteristics of stability of the symplectic algorithm for long time calculation of Hamiltonian systems with infi- nite dimensions. On the occasion of the commemoration of the 16th anniversary of Kang Feng’s death and the 89th anniversary of his birth, I think it is especially worthwhile to praise and promote what was embodied in the lifetime’s work of Kang Feng — “ indepen- dence in spirit, freedom in thinking”. 1 Now everyone is talking about scientific inno- vation, which needs a talented person to accomplish. What type of person is needed most? A person who is just a parrot or who has an “independent spirit, freely think- ing”? The conclusion is self-evident. Scientific innovation requires strong academic atmosphere. Is it determined by only one person or by all of the team members? This is also self-evident. From Kang Feng’s scientific career, we can easily find that the key to the problem of scientific innovation is “independence in spirit, freedom in thinking”, and that needs to be allowed to develop and expand. Kang Feng had planned to write a monograph about a symplectic algorithm for Hamiltonian systems. He had accumulated some manuscripts, but failed to complete it because he died too early due to sickness. Fortunately, his students and Professor Mengzhao Qin (see the photo on the previous page), one of the early collaborators, spent 15 years and finally completed this book based on Kang Feng’s plan, realizing his wish. It is not only an authoritative exposition of this research field, but also an

1 Yinke engraved on a stele in 1929 in memory of Guowei in campus of Tsinghua University. xii Foreword exposure of the academic thought of a master of science, which gives an example of how an original and innovative scientific discovery is initiated and developed from beginning to end in China. We would also like to thank Zhejiang Science and Technology Publishing House, which made a great contribution to the Chinese scientific cause through the publication of this manuscript. Although Kang Feng died 16 years ago, his scientific legacy has been inherited and developed by the younger generation of scientists. His scientific spirit and thought still elicit care, thinking and resonance in us. He is still living in the hearts of us.

Duan, Feng Member of Chinese Academy of Sciences University Nanjing September 20, 2009 Preface

It has been 16 years since Kang Feng passed away. It is our honor to publish the En- glish version of Symplectic Algorithm for Hamiltonian Systems, so that more readers can see the history of the development of symplectic algorithms. In particular, after the death of Kang Feng, the development of symplectic algorithms became more so- phisticated and there have been a series of monographs published in this area, e.g., Sanz-Serna & M.P. Calvo’s Numerical Hamiltonian Problems published in 1994 by Chapman and Hall Publishing House; E. Hairer, C. Lubich and G. Wanner’s Geo- metrical Numerical Integration published in 2001 by Springer Verlag; B. Leimkuhler and S. Reich’s Simulating Hamiltonian Dynamics published in 2004 by Cambridge University Press. The symplectic algorithm has been developed from ordinary dif- ferential equations to partial differential equations, from a symplectic structure to a multi-symplectic structure. This is largely due to the promotion of this work by J. Marsden of the USA and T. Bridge and others in Britain. Starting with a symplectic structure, J. Marsden first developed the Lagrange symplectic structure, and then to the multi-symplectic structure. He finally proposed a symplectic structure that meets the requirement of the Lagrangian form from the variational principle by giving up the boundary conditions. On the other hand, T. Bridge and others used the multi- symplectic structure to derive directly the multi-symplectic Hamilton equations, and then constructed the difference schemes that preserve the symplectic structure in both time and space. Both methods can be regarded as equivalent in the algorithmic sense. Now, in this monograph, most of the content refers only to ordinary differential equations. Kang Feng and his algorithms research group working on the symplectic algorithm did some foundation work. In particular, I would like to point out three nega- tive theorems: “ non-existence of energy preserving scheme”, “ non-existence of mul- tistep linear symplectic scheme”, and “ non-existence of volume-preserving scheme form rational fraction expression”. In addition, generating function theory is not only rich in analytical mechanics and Hamilton–Jacobi equations. At the same time, the construction of symplectic schemes provides a tool for any order accuracy difference scheme. The formal power series proposed by Kang Feng had a profound impact on the later developed “ backward error series” work ,“ modified equation” and “ modified integrator”. The symplectic algorithm developed very quickly, soon to be extended to the ge- ometric method. The structure preserving algorithm (not only preserving the geomet- rical structure, but also the physical structure, etc.) preserves the algebraic structure to present the algorithm, and preserves the differential complex algorithm. Many other prominent people have contributed to the symplectic method in addition to those mentioned above. There are various methods related to structure preserving algorithms and for important contributions the readers are referred to R. McLach- lan & GRW Quispel “ Geometric integration for ODEs” and T. Bridges & S. Reich “ Numerical methods for Hamiltonian PDEs”. The book describes the symplectic geometric algorithms and theoretical basis for a number of related algorithms. Most of the contents are a collection of lectures given xiv Preface by Kang Feng at Beijing University. Most of other sections are a collection of papers which were written by group members. Compared to the previous Chinese version, the present English one has been im- proved in the following respects. First of all, to correct a number of errors and mis- takes contained in the Chinese version. Besides, parts of Chapter 1 and Chapter 2 were removed, while some new content was added to Chapter 4, Chapter 7, Chapter 8, Chapter 9 and Chapter 10. More importantly, four new chapters — Chapter 13 to Chapter 16 were added. Chapter 13 is devoted to the KAM theorem for the symplectic algorithm. We invited Professor Zaijiu Shang , a former PhD student of Kang Feng to compose this chapter. Chapter 14 is called Variational Integrator. This chapter re- flects the work of the Nobel Prize winner Professor Zhengdao who proposed in the 1980s to preserve the energy variational integrator, but had not explained at that time that it had a Lagrange symplectic type, which satisfied the Lagrange symplectic structure. Together with J. Marsden he proposed the variational integrator trail con- nection, which leads from the variational integrator. Just like J. Marsden, he hoped this can link up with the finite element method. Chapter 15 is about Birkhoffian Sys- tems, describing a class of dissipative structures for Birkohoffian systems to preserve the dissipation of the Birkhoff structure. Chapter 16 is devoted to Multisymplectic and Variational Integrators, providing a summary of the widespread applications of multisymplectic integrators in the infinitely dimensional Hamiltonian systems. We would also like to thank every member of the Kang Feng’s research group for symplectic algorithms: Huamo , Daoliu Wang, Zaijiu Shang, Yifa Tang, Jialin Hong, Wangyao Li, Min Chen, Shuanghu Wang, Pingfu , Jingbo Chen, Yushun Wang, Yajuan , Hongwei Li, Jianqiang Sun, Tingting Liu, Hongling , Yimin ; and those who have been to the USA: Ge, Chunwang Li, Yuhua Wu, Meiqing , Wenjie , Shengtai Li, Lixin Jiang, and Haibin Shu. They made contributions to the symplectic algorithm over different periods of time. The authors would also like to thank the National Natural Science Foundation, the National Climbing Program projects, and the State’s Key Basic Research Projects for their financial support. Finally, the authors would also like to thank the Mathematics and Systems Science Research Institute of the Chinese Academy of Sciences, the Computational Mathematics and Computational Science and Engineering Institute, and the State Key Laboratory of Computational Science and Engineering for their support. The editors of this book have received help from E. Hairer, who provided a tem- plate from Springer publishing house. I would also like to thank F. Holzwarth at Springer publishing house and Linbo Zhang of our institute, and others who helped me successfully publish this book. For the English translation, I thank Dr. Shengtai Li for comprehensive proof- reading and polishing, and the editing of Miss . For the English version of the publication I would also like to thank the help of the Chinese Academy of Sciences Institute of Mathematics. Because Kang Feng has passed away, it may not be possible to provide a comprehensive representation of his academic thought, and the book will inevitably contain some errors. I accept the responsibility for any errors and welcome criticism and corrections. Preface xv

We would also like to thank Springer Beijing Representation Office and Zhejiang Science and Technology Publishing House, which made a great contribution to the Chinese scientific cause through the publication of this manuscript. We are especially grateful to thank Lisa Fan, W. Y. , L. L. Liu and X. M. for carefully reading and finding some misprints, wrong signs and other mistakes. This book is supported by National Natural Science Foundation of China under grant No.G10871099 ; supported by the Project of National 863 Plan of China (grant No.2006AA09A102-08); and supported by the National Basic Research Program of China (973 Program) (Grant No. 2007CB209603).

Mengzhao Qin Institute of Computational Mathematics and Scientific Engineering Computing Beijing September 20, 2009 Contents

Introduction ...... 1 Bibliography ...... 35

1. Preliminaries of Differentiable Manifolds ...... 39 1.1 Differentiable Manifolds ...... 40 1.1.1 Differentiable Manifolds and Differentiable Mapping . . . . 40 1.1.2 Tangent Space and Differentials ...... 43 1.1.3 Submanifolds ...... 46 1.1.4 Submersion and Transversal ...... 51 1.2 Tangent Bundle ...... 56 1.2.1 Tangent Bundle and Orientation ...... 56 1.2.2 Vector Field and Flow ...... 62 1.3 Exterior Product ...... 64 1.3.1 Exterior Form ...... 66 1.3.2 Exterior Algebra...... 68 1.4 Foundation of Differential Form ...... 75 1.4.1 Differential Form ...... 76 1.4.2 The Behavior of Differential Forms under Maps ...... 80 1.4.3 Exterior Differential...... 82 1.4.4 Poincare´ Lemma and Its Inverse Lemma...... 84 1.4.5 Differential Form in R3 ...... 86 1.4.6 Hodge Duality and Star Operators ...... 88 1.4.7 Codifferential Operator δ ...... 89 1.4.8 Laplace–Beltrami Operator ...... 90 1.5 Integration on a Manifold ...... 91 1.5.1 Geometrical Preliminary ...... 91 1.5.2 Integration and Stokes Theorem ...... 93 1.5.3 Some Classical Theories on Vector Analysis ...... 96 1.6 Cohomology and Homology ...... 98 1.7 Lie Derivative...... 99 1.7.1 Vector Fields as Differential Operator ...... 99 1.7.2 Flows of Vector Fields ...... 101 1.7.3 Lie Derivative and Contraction ...... 103 Bibliography ...... 111 xviii Contents

2. Symplectic Algebra and Geometry Preliminaries ...... 113 2.1 Symplectic Algebra and Orthogonal Algebra ...... 113 2.1.1 Bilinear Form ...... 113 2.1.2 Sesquilinear Form ...... 116 2.1.3 Scalar Product, Hermitian Product ...... 117 2.1.4 Invariant Groups for Scalar Products ...... 119 2.1.5 Real Representation of Complex Vector Space ...... 121 2.1.6 Complexification of Real Vector Space and Real Linear Transformation ...... 123 2.1.7 Lie Algebra for GL(n, F) ...... 124 2.2 Canonical Reductions of Bilinear Forms ...... 128 2.2.1 Congruent Reductions ...... 129 2.2.2 Congruence Canonical Forms of Conformally Symmet- ric and Hermitian Matrices ...... 130 2.2.3 Similar Reduction to Canonical Forms under Orthogo- nal Transformation ...... 134 2.3 Symplectic Space...... 137 2.3.1 Symplectic Space and Its Subspace ...... 137 2.3.2 Symplectic Group ...... 144 2.3.3 Lagrangian Subspaces ...... 147 2.3.4 Special Types of Sp(2n) ...... 148 2.3.5 Generators of Sp(2n) ...... 155 2.3.6 Eigenvalues of Symplectic and Infinitesimal Matrices . . . 158 2.3.7 Generating Functions for Lagrangian Subspaces ...... 160 2.3.8 Generalized Lagrangian Subspaces ...... 162 Bibliography ...... 164

3. Hamiltonian Mechanics and Symplectic Geometry ...... 165 3.1 Symplectic Manifold ...... 165 3.1.1 Symplectic Structure on Manifolds ...... 165 3.1.2 Standard Symplectic Structure on Cotangent Bundles . . . . 166 3.1.3 Hamiltonian Vector Fields ...... 167 3.1.4 Darboux Theorem ...... 168 3.2 Hamiltonian Mechanics on R2n ...... 168 3.2.1 Phase Space on R2n and Canonical Systems ...... 169 3.2.2 Canonical Transformation ...... 172 3.2.3 Poisson Bracket ...... 177 3.2.4 Generating Functions...... 180 3.2.5 Hamilton–Jacobi Equations ...... 182 Bibliography ...... 185

4. Symplectic Difference Schemes for Hamiltonian Systems...... 187 4.1 Background ...... 187 4.1.1 Element and Notation for Hamiltonian Mechanics ...... 187 Contents xix

4.1.2 Geometrical Meaning of Preserving Symplectic Struc- ture ω ...... 189 4.1.3 Some Properties of a Symplectic Matrix ...... 190 4.2 Symplectic Schemes for Linear Hamiltonian Systems ...... 192 4.2.1 Some Symplectic Schemes for Linear Hamiltonian Sys- tems ...... 192 4.2.2 Symplectic Schemes Based on Pade´ Approximation .....193 4.2.3 Generalized Cayley Transformation and Its Application . . 197 4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian Sys- tem ...... 200 4.4 Explicit Symplectic Scheme for Hamiltonian System ...... 203 4.4.1 Systems with Nilpotent of Degree 2 ...... 204 4.4.2 Symplectically Separable Hamiltonian Systems...... 205 4.4.3 Separability of All Polynomials in R2n ...... 207 4.5 Energy-conservative Schemes by Hamiltonian Difference ...... 209 Bibliography ...... 211

5. The Generating Function Method ...... 213 5.1 Linear Fractional Transformation ...... 213 5.2 Symplectic, Gradient Mapping and Generating Function ...... 215 5.3 Generating Functions for the Phase Flow ...... 221 5.4 Construction of Canonical Difference Schemes ...... 226 5.5 Further Remarks on Generating Function ...... 231 5.6 Conservation Laws ...... 234 5.7 Convergence of Symplectic Difference Schemes ...... 239 5.8 Symplectic Schemes for Nonautonomous System ...... 242 Bibliography ...... 247

6. The Calculus of Generating Functions and Formal Energy ...... 249 6.1 Darboux Transformation ...... 249 6.2 Normalization of Darboux Transformation ...... 251 6.3 Transform Properties of Generator Maps and Generating Functions 255 6.4 Invariance of Generating Functions and Commutativity of Gener- ator Maps ...... 261 6.5 Formal Energy for Hamiltonian Algorithm ...... 264 6.6 Ge–Marsden Theorem...... 273 Bibliography ...... 275

7. Symplectic Runge–Kutta Methods ...... 277 7.1 Multistage Symplectic Runge–Kutta Method ...... 277 7.1.1 Definition and Properties of Symplectic R–K Method. . . . 277 7.1.2 Symplectic Conditions for R–K Method ...... 281 7.1.3 Diagonally Implicit Symplectic R–K Method ...... 284 7.1.4 Rooted Tree Theory ...... 288 7.1.5 Simplified Conditions for Symplectic R–K Method .....297 xx Contents

7.2 Symplectic P–R–K Method ...... 302 7.2.1 P–R–K Method ...... 302 7.2.2 Symplified Order Conditions of Explicit Symplectic R–K Method ...... 307 7.3 Symplectic R–K–N Method ...... 319 7.3.1 Order Conditions for Symplectic R–K–N Method ...... 319 7.3.2 The 3-Stage and 4-th order Symplectic R–K–N Method . 323 7.3.3 Symplified Order Conditions for Symplectic R–K–N Method ...... 327 7.4 Formal Energy for Symplectic R–K Method ...... 333 7.4.1 Modified Equation ...... 334 7.4.2 Formal Energy for Symplectic R–K Method ...... 339 7.5 Definition of a(t) and b(t) ...... 345 7.5.1 Centered Euler Scheme ...... 345 7.5.2 Gauss–Legendre Method ...... 346 7.5.3 Diagonal Implicit R–K Method ...... 347 7.6 Multistep Symplectic Method ...... 347 7.6.1 Linear Multistep Method...... 347 7.6.2 Symplectic LMM for Linear Hamiltonian Systems ...... 348 7.6.3 Rational Approximations to Exp and Log Function ...... 352 Bibliography ...... 357

8. Composition Scheme ...... 365 8.1 Construction of Fourth Order with 3-Stage Scheme ...... 365 8.1.1 For Single Equation ...... 365 8.1.2 For System of Equations ...... 370 8.2 Adjoint Method and Self-Adjoint Method ...... 372 8.3 Construction of Higher Order Schemes ...... 377 8.4 Stability Analysis for Composition Scheme ...... 388 8.5 Application of Composition Schemes to PDE ...... 396 8.6 H-Stability of Hamiltonian System ...... 401 Bibliography ...... 405

9. Formal Power Series and B-Series ...... 407 9.1 Notation ...... 407 9.2 Near-0 and Near-1 Formal Power Series ...... 409 9.3 Algorithmic Approximations to Phase Flows ...... 414 9.3.1 Approximations of Phase Flows and Numerical Method . 414 9.3.2 Typical Algorithm and Step Transition Map ...... 415 9.4 Related B-Series Works ...... 417 9.4.1 The Composition Laws ...... 418 9.4.2 Substitution Law ...... 432 9.4.3 The Logarithmic Map ...... 434 Bibliography ...... 441 Contents xxi

10. Volume-Preserving Methods for Source-Free Systems ...... 443 10.1 Liouville’s Theorem ...... 443 10.2 Volume-Preserving Schemes ...... 444 10.2.1 Conditions for Centered Euler Method to be Volume Preserving ...... 444 10.2.2 Separable Systems and Volume-PreservingExplicit Meth- ods...... 447 10.3 Source-Free System...... 449 10.4 Obstruction to Analytic Methods ...... 450 10.5 Decompositions of Source-Free Vector Fields ...... 452 10.6 Construction of Volume-Preserving Schemes...... 454 10.7 Some Special Discussions for Separable Source-Free Systems . . . . 458 10.8 Construction of Volume-PreservingScheme via Generating Func- tion ...... 460 10.8.1 Fundamental Theorem ...... 460 10.8.2 Construction of Volume-Preserving Schemes ...... 464 10.9 Some Volume-Preserving Algorithms ...... 467 10.9.1 Volume-Preserving R–K Methods ...... 467 10.9.2 Volume-Preserving 2-Stage P–R–K Methods ...... 471 10.9.3 Some Generalizations ...... 473 10.9.4 Some Explanations ...... 474 Bibliography ...... 476

11. Contact Algorithms for Contact Dynamical Systems ...... 477 11.1 Contact Structure ...... 477 11.1.1 Basic Concepts of Contact Geometry ...... 477 11.1.2 Contact Structure ...... 480 11.2 Contactization and Symplectization ...... 484 11.3 Contact Generating Functions for Contact Maps ...... 488 11.4 Contact Algorithms for Contact Systems ...... 492 11.4.1 Q Contact Algorithm ...... 493 11.4.2 P Contact Algorithm ...... 493 11.4.3 C Contact Algorithm ...... 493 11.5 Hamilton–Jacobi Equations for Contact Systems ...... 494 Bibliography ...... 497

12. Poisson Bracket and Lie–Poisson Schemes ...... 499 12.1 Poisson Bracket and Lie–Poisson Systems ...... 499 12.1.1 Poisson Bracket ...... 499 12.1.2 Lie–Poisson Systems ...... 501 12.1.3 Introduction of the Generalized Rigid Body Motion .....505 12.2 Constructing Difference Schemes for Linear Poisson Systems . . . . 507 12.2.1 Constructing Difference Schemes for Linear Poisson Systems ...... 508 xxii Contents

12.2.2 Construction of Difference Schemes for General Pois- son Manifold ...... 509 12.2.3 Answers of Some Questions ...... 511 12.3 Generating Function and Lie–Poisson Scheme ...... 514 12.3.1 Lie–Poisson–Hamilton–Jacobi (LPHJ) Equation and Gen- erating Function ...... 514 12.3.2 Construction of Lie–Poisson Schemes via Generating Function ...... 519 12.4 Construction of Structure Preserving Schemes for Rigid Body . . . . 523 12.4.1 Rigid Body in Euclidean Space ...... 523 12.4.2 Energy-Preserving and Angular Momentum-Preserving Schemes for Rigid Body ...... 525 12.4.3 Orbit-Preserving and Angular-Momentum-Preserving Ex- plicit Scheme ...... 527 12.4.4 Lie–Poisson Schemes for Free Rigid Body ...... 530 12.4.5 Lie–Poisson Scheme on Heavy Top ...... 535 12.4.6 Other Lie–Poisson Algorithm ...... 538 12.5 Relation Among Some Special Group and Its Lie Algebra ...... 543 12.5.1 Relation Among SO(3),so(3) and SH1,SU(2) ...... 543 12.5.2 Representations of Some Functions in SO(3) ...... 545 Bibliography ...... 547

13. KAM Theorem of Symplectic Algorithms ...... 549 13.1 Brief Introduction to Stability of Geometric Numerical Algorithms 549 13.2 Mapping Version of the KAM Theorem ...... 551 13.2.1 Formulation of the Theorem ...... 551 13.2.2 Outline of the Proof of the Theorems ...... 554 13.2.3 Application to Small Twist Mappings ...... 558 13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems559 13.3.1 Symplectic Algorithms as Small Twist Mappings ...... 560 13.3.2 Numerical Version of KAM Theorem ...... 564 13.4 Resonant and Diophantine Step Sizes ...... 568 13.4.1 Step Size Resonance ...... 568 13.4.2 Diophantine Step Sizes ...... 569 13.4.3 Invariant Tori and Further Remarks ...... 574 Bibliography ...... 578

14. Lee-Variational Integrator ...... 581 14.1 Total Variation in Lagrangian Formalism ...... 581 14.1.1 Variational Principle in Lagrangian Mechanics ...... 581 14.1.2 Total Variation for Lagrangian Mechanics ...... 583 14.1.3 Discrete Mechanics and Variational Integrators ...... 586 14.1.4 Concluding Remarks ...... 591 14.2 Total Variation in Hamiltonian Formalism ...... 591 14.2.1 Variational Principle in Hamiltonian Mechanics ...... 591 Contents xxiii

14.2.2 Total Variation in Hamiltonian Mechanics ...... 593 14.2.3 Symplectic-Energy Integrators ...... 596 14.2.4 High Order Symplectic-Energy Integrator ...... 600 14.2.5 An Example and an Optimization Method ...... 603 14.2.6 Concluding Remarks ...... 605 14.3 Discrete Mechanics Based on Finite Element Methods...... 606 14.3.1 Discrete Mechanics Based on Linear Finite Element .....606 14.3.2 Discrete Mechanics with Lagrangian of High Order .....608 14.3.3 Time Steps as Variables...... 613 14.3.4 Conclusions ...... 614 Bibliography ...... 615

15. Structure Preserving Schemes for Birkhoff Systems ...... 617 15.1 Introduction ...... 617 15.2 Birkhoffian Systems ...... 618 15.3 Generating Functions for K(z,t)-Symplectic Mappings ...... 621 15.4 Symplectic Difference Schemes for Birkhoffian Systems ...... 625 15.5 Example ...... 629 15.6 Numerical Experiments ...... 634 Bibliography ...... 639

16. Multisymplectic and Variational Integrators ...... 641 16.1 Introduction ...... 641 16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Sys- tems ...... 642 16.3 Multisymplectic Integrators and Composition Methods ...... 646 16.4 Variational Integrators ...... 652 16.5 Some Generalizations ...... 654 Bibliography ...... 658

Symbol ...... 663

Index ...... 669