Geometric numerical integration of simple dynamical systems

Abstract

Understanding the behavior of a dynamical system is usually accomplished by visualization of its phase space portraits. Finite element simulations of dynamical systems yield a very high dimensionality of phase space, i.e. twice the number of nodal degrees of freedom. Therefore insight into phase space structure can only be gained by reduction of the model's dimensionality. The phase space of Hamiltonian systems is of particular interest because of its inherent geometric features namely being the co-tangent bundle of the configuration space of the problem and therefore having a natural symplectic structure. In this contribution a class of geometry preserving integrators based on Lie-groups and -algebras is presented which preserve these geometric features exactly. Examples of calculations for a simple dynamical system are detailed.