Featured Review: Geometric Numerical Integration: Structure-Preserving Al

Geometric integration is a field that was invented “too late,” and this book is an impressive record of a decade of catching up. It could have been invented at any time since Newton.1 It could have been invented in 1957, with “a marvellous paper, short, clear, elegant, written in one week, submitted for publication—and never published” [4]. It could have been invented in 1967, when Loup Verlet introduced the leapfrog (Störmer/Verlet) method to molecular dynamics; its many desirable geometric properties were not appreciated for many years. In 1983 it was written that the symplecticity of leapfrog was well known, although perhaps it was only truly well known at SLAC and Los Alamos, and not known at all elsewhere. When I was a grad student in the late 1980s, photocopies of [3] were passed around and created a great stir, especially its Figure 2, which compared a new, symplectic integrator (showing a crisp, well-defined curve) to the hapless classical Runge–Kutta (a shapeless blur). Perhaps this mysterious word “symplectic” was also part of the appeal—some kind of secret weapon was in play. Of course, we students had no idea what was going on, and such comparisons were later criticized as unfair, but in any event, [3] unleashed the floodgates, and the 1990s saw a period of rapid development, first of symplectic integrators and then of more general geometric integrators. It seems to me that in some areas there is a gulf between the theoreticians and the practitioners of numerical methods. Each group ignores the methods and experience of the other. Without wanting to make too much of it, geometric integration does seem to be such an area. For decades numerical analysts were either ignorant or skeptical of methods (such as leapfrog) developed and used to great effect in molecular dynamics, quantum mechanics, and elsewhere, perhaps because the methods didn’t look good by the traditional criteria based on small and controllable local errors. This situation can only make it harder for numerical analysts to “sell” any new methods they may develop to their potential “customers,” which is, after all, the whole point of developing the methods. Let us have a closer look at this leapfrog method. In [4], Hairer, Lubich, and Wanner offered a short preview of the book currently under review, exclusively from the point of view of this method. We start with a mechanical system with configuration space R (for example, a system of n/3 particles moving in three dimensions) and phase space R2n. Suppose the system has potential energy V (q) and kinetic energy T (p) = 2‖p‖2, so that its total energy, or Hamiltonian, is H(q, p) = T (p)+V (q). The