Arne Marthinsen

During the last few years, different approaches for integrating ordinary differential equations on manifolds have been published. In this work, we consider two of these approaches. We present some numerical experiments showing benefits and some pitfalls when using the new methods. To demonstrate how they work, we compare with well known classical methods, e.g. Newmark and Runge-Kutta methods. AMS Subject Classification: 65L05, 34A50 Key Words: ordinary differential equations, manifolds, numerical analysis, initial value problems. 1 Introduction In recent years, many authors have argued that certain problems in mechanics cannot be solved in a satisfactory way by classical numerical methods. For instance, it might be desirable that problems which possess certain invariants are solved with a numerical method that preserves the same invariants. Typical examples are the symplectic structure in a Hamiltonian system, the energy in a conservative mechanical system and the angular momentum o..