Solving Numerically Hamiltonian Systems

where / is a smooth function. The basic theory of numerical methods for (1) has been known for more than thirty years, see e.g. [8]. This theory, in tandem with practical experimentation, has led to the development of general software packages for the efficient solution of (1). It is perhaps remarkable that both the theory and the packages do not take into account any structure the problem may have and work under virtually no assumption on the (smooth) vector field / . This contributes to the elegance of the theory and to the versatility of the software. However, it is clear that a method that can solve "all" problems is bound to be inefficient in some problems. Stiff problems [9], frequent in many applications, provide an example of problems of the format (1) where general packages are very inefficient. Accordingly, a special theory and special software have been created to cope with stiff problems. Are there other classes of problems of the form (1) that deserve a separate study? In recent years much work has been done on special methods for Hamiltonian problems. Of course, Hamiltonian problems [11] play a crucial role as mathematical models of situations where dissipative effects are absent or may be ignored. Most special methods for Hamiltonian problems are symplectic methods; other possibilities, not discussed here, include reversible and energy-conserving methods [16]. Early references on symplectic integration are Channell [4], Feng [5], and Ruth [12]. In the last ten years the growth of the "symplectic" literature has been impressive, both in mathematics and in the various application fields. The monograph [16] contains over a hundred references from the mathematical literature. The second edition of the excellent treatise by Hairer, N0rsett, and Wanner [8] includes a section on symplectic integration.