Abstract
Trying to extend a local definition of a surface of a section, and the corresponding Poincare map to a global one, one can encounter severe difficulties. We show that global transverse sections often do not exist for Hamiltonian systems with two degrees of freedom. As a consequence we present a method to generate the so-called W-section, which by construction will be intersected by (almost) all orbits. Depending on the type of tangent set in the surface of the section, we distinguish five types of W-sections. The method is illustrated by a number of examples, most notably the quartic potential and the double pendulum. W-sections can also be applied to higher dimensional Hamiltonian systems and to dissipative systems.