Regular and Chaotic Motion in Hamiltonian Systems Harry Varvoglis 1 Section of Astrophysics Astronomy & Mechanics, Department of Physics, Aristotle University of Thessaloniki, 541 24 Thessaloniki, Greece (varvogli@physics.auth.gr) 1 Introduction All laws that describe the time evolution of a continuous system are given in the form of differential equations, ordinary (if the law involves one inde- pendent variable) or partial (if the law involves two or more independent variables). Historically the first law of this type was Newton’s second law of motion. Since then Dynamics, as it is customary to name the branch of Mechanics that studies the motion of a body as the result of a force acting on it, has become the “typical” case that comes into one’s mind when a system of ordinary differential equation is given, although this system might as well describe any other system, e.g. physical, chemical, biological, financial etc. In particular the study of “conservative” dynamical systems, i.e. systems of ordinary differential equations that originate from a time-independent Hamil- tonian function, has become a thoroughly developed area, because of the fact that mechanical energy is very often conserved, although many other physi- cal phenomena, beyond motion, can be described by Hamiltonian systems as well. In what follows we will restrict ourselves exactly to the study of Hamil- tonian systems, as typical dynamical systems that find applications in many scientific disciplines. The method used traditionally in the study of a conservative dynamical sys- tem, whose Hamiltonian leads to a complex set of differential equations, is to follow a sequence of approximations. In the beginning, we try to solve exactly the equations derived from a simplified form of the Hamiltonian, the so-called zero-order approximation, omitting complex or non-linear terms. Then we revert to the full Hamiltonian and consider the omitted terms as a “perturbation” of the zero order approximation. We then modify the initial solution by a sequence of “corrections”, which are expected to describe better and better the real system. This method is based on the implicit assumption that the sequence of suc- cessive approximations converges to the “real” solution. Since the method is based on the fact that the solution of the zero order dynamical system is known exactly, it is closely connected to the theory of integrable, or else regular dynamical systems. By this term, we denote the Hamiltonian dynam- ical systems whose differential equations of motion can be solved at least by quadratures, i.e. they can be expressed as integrals of functions of one