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Simulating Hamiltonian dynamics. (English) Zbl 1069.65139

Cambridge Monographs on Applied and Computational Mathematics 14. Cambridge: Cambridge University Press (ISBN 0-521-77290-7/hbk; 978-0-511-61411-8/ebook). xvi, 379 p. (2004).
Geometric integrators are numerical methods that preserve relevant properties of the flow of the underlying differential equations. The book under review is concerned with geometric integrators for the class of conservative Hamiltonian systems.
Due to their application in mathematical modelling, geometric integration has been a very active focus of research in the last two decades and a number of important theoretical results in parallel with the development of a vast computational experience has expanded considerably the knowledge on this subject and several monographs have been published in the last decade.
An excellent state of the art on symplectic integration is contained in the book by J. M. Sanz-Serna and M. P. Calvo [Numerical Hamiltonian problems (1994; Zbl 0816.65042)]. More recently, the monograph of E. Hairer, C. Lubich and G. Wanner [Geometric numerical integration (2002; Zbl 0994.65135)] provided, with the well known style of the authors, a rigourous theoretical foundation of the subject. The book under review of Leimkuhler and Reich is a new addition to the existing literature with a different point of view.
First of all, the book is mainly intended to be used as a textbook in courses of computational mechanics: Some exercises are included at each chapter to ensure the student the understanding of the theory and also numerical examples are presented to check and validate the theory. Secondly, the authors do not have attempted to give a comprehensive treatment of the subject and they have studied only those numerical methods that are used in practical applications.
The book contains twelve chapters and (in opinion of this reviewer) consist of three main parts. A short overview of their content may be appropriate. In the first part, Chapters 1 to 3, after introducing several Hamiltonian problems to illustrate the class of problems, some background material of one step methods for differential equations and Hamiltonian dynamics has been included to make the book available for a wide audience. Although this material can be found in classical texts of numerical methods for differential equations and classical mechanics, my main criticism is the lack of precision at some points. Thus, in p. 38, equation (3.7) defines a Hamiltonian system as a \(2d-\)dimensional differential system in \( \mathbf{z}=(\mathbf{p},\mathbf{q})^T \in\mathbb{R}^{2d},\) with the special form \[ {d \over dt} {\mathbf{z}} = J \; \nabla_{\mathbf{z}} H({\mathbf{z}}), \qquad J= \left( \begin{matrix} 0 & I_d \\ -I_d & 0\end{matrix}\right), \tag{1} \] where \( J \in\mathbb{R}^{2d \times 2d}\) is the canonical structure matrix and \( H = H({\mathbf{z}})\) the Hamiltonian function. Then it is said: “the term canonical is reserved for Hamiltonian systems with \(J\) as in (1), but Hamiltonian systems can be generalized in various ways without altering the discussion of their geometric properties in an essential way...”. In view of this the reader might think that the term “canonical Hamiltonian system” is reserved exclusively to (1) whereas “Hamiltonian system” is for generalized versions (such as Poisson systems ). However in the remainder of the book the word “ canonical” is often skipped.
The second part of the book starts in Chapter 4 with the study of geometric integrators. After introducing the concept of symplectic integrator it is proved that three basic integrators: the so called asymmetric Euler (A and B) methods (also referred to in the literature as symplectic Euler methods) and the implicit midpoint rule are symplectic. The technique of Hamiltonian splitting is presented here as the main approach to construct symplectic methods for canonical Hamiltonian systems whose Hamiltonian can be decomposed as a sum of integrable Hamiltonians (e.g. separable). Also symmetric compositions are introduced to derive symplectic methods and some symplectic methods as the mid point rule and the Störmer-Verlet method. The preservation of other properties such as time-reversal symmetry and first integrals is also considered. The Chapter ends with numerical experiments illustrating the behaviour of some methods for two problems. Chapter 5 is devoted to the backward error analysis of the numerical integrators under consideration. Since this technique allows us to study the qualitative behaviour of the integrators it plays an essential role in the study of structure preservation of geometric integrators. The treatment of the subject is fair complete, including recent researches of one of the authors of the book. Some numerical examples are presented to illustrate the theory.
Chapter 6 is concerned with the construction of higher order symplectic methods. In the first part the authors study the technique of (symmetric) compositions of low order methods that allows to construct explicit higher order methods for separable Hamiltonians. A reference to the post processing technique is also included. In the second part of this Chapter the authors survey the well known theory of symplectic Runge-Kutta and partitioned Runge-Kutta methods for general Hamiltonians.
Chapters 7 and 8 deal with constrained mechanical systems, i.e. systems described by \(n\) coordinates \( (q_1, \ldots,q_n)^T={\mathbf q} \), under conservative forces that derive from a potential function \( V=V( {\mathbf q})\), with the coordinates satisfying the holonomic constraints \( g_i({\mathbf q})=0, i=1, \ldots,m<n\). Such a system may be described by a set of Lagrangian differential-algebraic equations of type \[ {\mathbf M} { d^2 {\mathbf q} \over dt}= - \nabla_{\mathbf q} V({\mathbf q}) - \sum_{i=1}^m \nabla_{\mathbf q} g_i({\mathbf q}) \lambda_i, \quad g_i({\mathbf q})=0, \; (i=1, \ldots,m), \tag{2} \] where \( \lambda_i\) are the Lagrange multipliers and \({\mathbf M}\) the positive definite mass matrix. Although one could obtain explicitly the multipliers by differentiations of the constraints and eliminate them in the differential equations, in practice it is usually preferred to use a direct discretization of (2) preserving the constraints. In Chapter 7, the so called RATTLE and SHAKE direct methods are introduced and their equivalence and implementation are studied in detail. Further, a Hamiltonian formulation of constrained systems is introduced and a second order constraint preserving symplectic method is given. A nice example of a constrained system is the motion of a rigid body and Chapter 8 deals with this problem. Here the authors present a complete derivation of the constrained Lagrangian and Hamiltonian equations of a rigid body in terms of the coordinates of the center of mass and the rotation matrix. The results of several numerical experiments with the above methods are given and analyzed. Further, several alternative descriptions of the rigid body problem with other sets of variables are also considered.
Chapter 9 deals with the use of variable step size in geometric integrators. As a general rule, adaptive step size improves considerably the efficiency of numerical integrators but usually destroys the preservation properties and the long term behaviour of symplectic integrators. Thus, in general, the advantages of adaptivity in geometric integration for the explicit integrators considered in the book are not clear. Nevertheless, the authors have collected a number of approaches proposed in the literature, such as the isoenergetic regularization of Poincaré, to introduce variable step size while maintaining some properties of the flow and presented numerical experiments to test their behaviour.
The last part, Chapters 10 to 12, is devoted to three important problems of geometric integration due to their practical applications: Highly oscillatory problems, molecular dynamics and Hamiltonian partial differential equations. These are more specialized problems in which the authors of the book have made relevant contributions in the last years and consequently these chapters reflect their experience in these subjects. A specialized reader interested in applications will find this part of most value because it presents in a unified way many techniques that have been proposed recently in the literature. However for a course in computational mechanics a careful selection of material with additional explanations would be required.
In summary, this new book on geometric integration of Hamiltonian systems is a valuable addition to the subject that may be very useful not only as textbook for courses in computational dynamics but also for researchers in the design of effective integrators in molecular dynamics and other areas of applied mathematics because includes most of the recent research in the subject.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
70E55 Dynamics of multibody systems
70F10 \(n\)-body problems
70F20 Holonomic systems related to the dynamics of a system of particles
70H05 Hamilton’s equations
70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
37N05 Dynamical systems in classical and celestial mechanics
37N15 Dynamical systems in solid mechanics
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
37Jxx Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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