Bøger i Theoretical and Mathematical Physics serien

The main aim of this book is to develop the methods of Global Analysis and of Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as quite distant from each other and requiring different methods of investigation. Among those areas we mention classical mechanics on non-linear configuration spaces, some problems of statistical and quantum physics, hydrodynamics, etc. The idea, yielding the unification of these topics, is based on the use of a geometrically invariant form of Newton's second law and its analogs (stochastic, set-valued, infinite-dimensional, etc.) as a fundamental equation of motion.

Theoretical physics deals with physical models. The main requirements for a good physical model are simplicity and universality. Universal models which can be applied to describe a variety of different phenomena are very rare in physics and, therefore, they are of key importance. Such models attract the special attention of researchers as they can be used to describe underlying physical concepts in a simple way. Such models appear again and again over the years and in various forms, thus extending their applicability and educa­ tional value. The simplest example of this kind is the model of a pendulum; this universal model serves as a paradigm which encompasses basic features of various physical systems, and appears in many problems of very different physical context. Solids are usually described by complex models with many degrees of freedom and, therefore, the corresponding microscopic equations are rather complicated. However, over the years a relatively simple model, known these days as the Prenkel-K ontorova model, has become one of the fundamental and universal tools of low-dimensional nonlinear physics; this model describes a chain of classical particles coupled to their neighbors and subjected to a pe­ riodic on-site potential.