Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

Springer 1 Introduction This book reports on an unconventional explanation of the origin of chaos in Hamiltonian dynamics and on a new theory of the origin of thermodynamic phase transitions. The mathematical concepts and methods used are borrowed from Riemannian geometry and from elementary differential topology, respectively. The new approach proposed also unveils deep connections between the two mentioned topics. Written as a monograph on a new theoretical framework, this book is aimed at stimulating the active interest of both mathematicians and physicists in the many still open problems and potential applications of the theory discussed here. Thus we shall focus only on those particular aspects of the subjects treated that are necessary to follow the main conceptual construction of this volume. Many topics that would naturally find their place in a textbook, despite their general relevance will not be touched on if they are not necessary for following the leitmotif of the book. In order to ease the reading of the volume and to allow the reader to choose where to concentrate his attention, this introduction is written as a recapitulation of the content of the book, giving emphasis to the logical and conceptual development of the subjects tackled, and drawing attention to the main results (equations and formulas) worked out throughout the text. In this book, we shall consider physical systems described by N degrees of freedom (particles, classical spins, quasi-particles such as phonons, and so on), confined in a finite volume (therein free to move, or defined on a lattice), whose Hamiltonian is of the form H = 1 2 N i=1 p 2 i + V which we call " standard, " where the q's and the p's are, respectively, the coordinates and the conjugate momenta of the system. Our emphasis is on 2 1 Introduction