Simulating Hamiltonian Dynamics B. Leimkuhler and S. Reich Cambridge University Press (2005) 396 pp., price USD 75.00 [60.00 electronic] ISBN:0521772907
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.

Level: basic.

Numerical analysts know and know well that different types of differential equations require different numerical methods to obtain optimal results. One of the most interesting examples of this principle is the very active area of geometric integrators, or time-stepping methods which exactly conserve an energy, a symmetry, or some other conserved quantity implicit in the equations. This is a fascinating subject, as it entwines classical numerical analysis with geometry and dynamics.

This volume serves as a textbook for integrators which converse the symplecticness of a Hamiltonian system. (The term `symplecticness' seems, curiously, to be used only within the numerical analysis community and not within symplectic geometry.)

The authors are certainly correct in asserting that good numerical methods are invaluable in understanding chaotic dynamics, and that conservative systems demand a unique approach. The authors are less convincing in their prefatory assertion that the need for simulation is driven by a lack of mathematics to deal with chaotic dynamics: `...the new mathematics that would be needed to answer even the most basic questions regarding chaotic systems is still in its infancy. In the absence of a useful general theoretic method for analyzing complex nonlinear phenomena, simulation is increasingly pushed to the fore.' That might have been an appropriate line to pen a quarter century ago. Fortunately, the ideas in the remainder of the book are not so dated.

In order to cover the material on symplectic integrators for Hamiltonian dynamics, one must explain what is meant by the terms symplectic and Hamiltonian, thus necessitating an excursion to differential forms and Hamiltonian dynamics in Chapter 3 of the text. [Aside: While there is no lack of books on differential forms and their applications to Hamiltonian dynamics, there is a lack of clear, rigorous, geometric explanations in which authors resist the temptation to plunge the reader into a purgatory of notation. Arnold's MMCM is still the best source for learning about forms.] Hence, the present authors' need to try and explain differential forms, symplectic geometry, and Hamiltonian dynamics in thirty pages. It is a reasonable explanation, saved by a few well-crafted and well-placed figures. This may seem a picky and minor point --- after all, the book is all about the numerical methods. And, indeed, for students who are well-prepared to begin learning about symplectic integrators, the defects in Chapter 3 will not impede. Those students who come to learn the subject ex nihilo will likely find the notion of a symplectic map --- truly as beautiful and `organic' mathematical object as ever there was --- obscured in notation and equations with boxes around them.

As an illustrative example, consider the first `boxed-in' equations on the second page of Chapter 3. This gives the Hamiltonian form of Newtonian Mechanics:

dq/dt = M^-1p

dp/dt = -Ñ_qV(q)

Undergraduates tend to view the stuff in the box the way that mathematicians view the stuff after Theorem: It points to the essential, core truth. The equations above are boxed; the Euler-Lagrange equation is not; the action integral is not; the definition of the Poisson bracket is not; et cetera.

The physical book itself could be improved. While the illustrations are crisp, the book's margins induce claustrophobia and the font is one that all mathematicians will recognize as that used in SLITEX silde presentations. This font is bearable at the text density of a talk slide and unbearable at the density of the printed page. This is likely not the authors' fault, but it nevertheless detracts from their work.

It serves as partial recompense for the quality of the physical text that there is an electronic version of the text available in Abode .pdf format. The price of this is twenty percent cheaper than the physical text. It is for the customer to decide if the longevity of a physical text (covering what is likely to be a rapidly-developing area of research) is worth fifteen bucks.

This text is suitable for an advanced undergraduate or beginning graduate course. The text begins at an elementary level, but covers material which is very current, e.g., multi-symplectic integrators and PDEs. The exercises are rather good, with a nice balance between analytical and numerical projects, the latter of which can be quite open-ended. For a text which is, essentially, a numerical methods text, there is a firm grounding in physical applications and physical reasoning, both in examples and in exercises.