Poincaré Surface of Sections, Mappings

Abstract

We consider a system with two degrees of freedom, which we describe in four-dimensional phase space. In this (finite) space we define an (oriented) two-dimensional surface. If we then consider the trajectory in phase space, we are interested primarily in its piercing points through this surface. This piercing can occur repeatedly in the same direction. If the motion of the trajectory is determined by the Hamiltonian equations, then the n + 1-th piercing point depends only on the nth. The Hamiltonian thus induces a mapping n → n + 1 in the “Poincaré surface of section” (P.S.S.). The mapping transforms points of the P.S.S. into other (or the same) points of the P.S.S. In the following we shall limit ourselves to autonomous Hamiltonian systems, ∂H/∂t = 0, so that because of the canonicity (Liouville’s theorem) the mapping is area-preserving (canonical mapping).