Numerical Analysis of Constrained Hamiltonian Systems and the Formal Theory of Differential Equations

We show how the formal theory of diierential equations provides a unifying framework for some aspects of constrained Hamiltonian systems and of the numerical analysis of diierential algebraic equations, respectively. This concerns especially the Dirac algorithm for the construction of all constraints and various index concepts for diierential algebraic equations. 1. Introduction Constrained Hamiltonian systems arise in many elds, e.g. in multi-body dynamics or molecular dynamics. As it is rarely possible to solve them analytically, their numerical integration is of great importance. Due to the existence of the constraints, the equations of motion form a diierential algebraic equation, i.e. a system comprising diierential and algebraic equations. The straightforward application of standard numerical methods to diierential algebraic equations is usually not possible. One reason is the existence of hidden constraints or integrability conditions. These are further algebraic equations satis-ed by any solution of the original system but not part of it. They make especially a consistent initialization rather diicult. Physicists have developed various methods to deal with constrained Hamiltonian systems, although they are usually more interested in their quantization than in numerical computations. The Dirac theory 13, 14] provided not only the rst solution but represents still one of the most important approaches. The Dirac algorithm constructs all hidden constraints in a simple manner. A geometric version based on diierential equations was presented in 28]. Various authors developed independently geometric frameworks for the treatment of diierential algebraic equations (or more generally implicit diierential equations) 32, 33, 46]. These include algorithms for the construction of all integrability conditions, as this is important for an existence and uniqueness theory. Despite the fact that mechanical systems with constraints represent one of the main sources for diierential algebraic equations, the relation between these formalisms and the Dirac theory has apparently never been studied in detail. The purpose of this article is to point out that the mentioned physical and numerical theories, respectively, are special cases of the general problem of completion of a non-normal system of diierential equations. First solutions of this problem,