Introduction

Abstract

We shall introduce the concepts of symplectic manifolds, symplectic mappings and Hamiltonian vector fields. It is not the intention to give a systematic treatment of the Hamiltonian formalism, because it is already presented in many books. Rather we shall ask some questions related to these concepts which recently lead to new phenomena and interesting open problems. The question: “What can be done with a symplectic mapping?” leads, for example, to new symplectic invariants different from the volume and discussed in detail in subsequent chapters. We shall illustrate that a seemingly very different and old problem originating in celestial mechanics is related to these invariants. Namely, prompted by the Poincaré recurrence theorem, we ask whether a compact energy surface of a Hamiltonian vector field possesses a periodic orbit. For the very special case of a convex hypersurface in ℝ²ⁿ, historically one of the landmarks in this qualitative problem of Hamiltonian systems, we shall give an existence proof in order to illustrate the so-called direct method of the calculus of variation. This classical method is in contrast to the more recent methods introduced in the following chapters in order to establish global periodic solutions. At the end of the introduction we shall illustrate without proofs the rich and intricate orbit structure to be expected near a given periodic orbit. The considerations are based on the local, nonlinear Birkhoff-invariants presented in detail.