Regular and Chaotic Motion in Hamiltonian Systems

Abstract

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 independent 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 Hamiltonian function, has become a thoroughly developed area, because of the fact that mechanical energy is very often conserved, although many other physical phenomena, beyond motion, can be described by Hamiltonian systems as well. In what follows we will restrict ourselves exactly to the study of Hamiltonian systems, as typical dynamical systems that find applications in many scientific disciplines.