### Abstract:

Roels [1] showed that on a two dimensional symplectic manifold, an arbitrary vector field can be locally decomposed into the sum of a gradient vector field and a Hamilton vector field. The Roels decomposition was extended to be applicable to compact even dimensional manifolds by Mendes and Duarte [2]. Some of the limitations of local decomposition are overcome by incorporating modern work on Hodge decomposition. This leads to a theorem which, in some cases, allows an arbitrary vector field on an even m-dimensional non-compact manifold to be decomposed into one gradient vector field and up to m-1 Hamiltonian vector fields. The method of decomposition is condensed into an algorithm which can be implemented using computer algebra. This decomposition is then applied to chaotic vector fields on non-compact manifolds [3]. This extended Roels decomposition is also compared to Helmholz decomposition in R 3 . The thesis shows how Legendre polynomials can be used to simplify the Helmholz decomposition in non-trivial cases. Finally, integral preserving iterators for both autonomous and non-autonomous first integrals are discussed [4]. The Hamilton vector fields which result from Roels' decomposition have their Hamiltonians as first integrals.