ABSTRACT
Several of the exciting developments in the Classical Mechanics of Conservative and Dissipative Systems are briefly reviewed here, at this summerschool on Statistical Mechanics, assuming only a rudimentary knowledge of some graduate course in Mechanics: It turns out that the phase space of most Hamiltonian- and many Dissipative-systems is dotted with ‘Chaotic’ Regions in which some properties of many orbits are as random as coin tosses, even though the system is deterministic. While statistical methods appear to be applicable to those chaotic regions they are incompatible with the very smooth regular, ‘quasi-periodic', behavior in other regions, of which there is an abundance as well. Hence ‘Ergodicity’ and the ‘Approach to Thermal Eąuilibrium’ do not hold for moot Hamiltonian system. Parts of those chaotic regions do survive under some dissipative perturbations but the regular regions do not: Instead of being ‘attracted’ to the origin (damping) or to a periodic orbit (‘limit cycle’), the orbits -from the previously regular region- may now be attracted to the remainder of a chaotic region. The motion on these so called ‘Strange Attractors’ can be chaotic, ergodic and even ‘mixing’. The transition from simple limit cycles to very complicated ones,toa regime with Strange Attractors has recently been derived. This ‘Period-Doubling’ Transition provides a model for the Onset of Turbulent Behavior.
