Książki

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.

Geometry from Dynamics, Classical and Quantum

Foreword: The birth and the long gestation of a project.- Some examples of linear and nonlinear physical systems and their dynamical equations.- Equations of the motion for evolution systems.- Linear systems with infinite degrees of freedom.- Constructing nonlinear systems out of linear ones.- The language of geometry and dynamical systems: the linearity paradigm.- Linear dynamical systems: The algebraic viewpoint.- From linear dynamical systems to vector fields.- Exterior differential calculus on linear spaces.- Exterior differential calculus on submanifolds.- A tensorial characterization of linear structures.- Partial linear structures: Vector bundles.- Covariant calculus.- Riemannian and Pseudo-Riemannian metrics on linear vector spaces.- Invariant geometric structures and the classical formulations of dynamics of Poisson, Jacobi, Hamilton and Lagrange.- Linear vector fields.- Additional invariant structures for linear vector fields.- Poisson structures.- The inverse problem for Poisson structures.- Symplectic structures.- Lagrangian structures.- Invariant Hermitean structures and the geometry of quantum systems.- Invariant Hermitean inner products.- Complex structures and complex exterior calculus.- Algebras associated with Hermitean structures.- The geometry of quantum dynamical evolution.- The Geometry of Quantum Mechanics and the GNS construction.- Alternative Hermitean structures for quantum systems.- Folding and unfolding Classical and Quantum systems.- Introduction: separable dynamics.- The geometrical description of reduction.- The algebraic description.- Reduction in Quantum Mechanics.- Integrable and superintegrable systems.- The geometrization of the notion of integrability.- The normal form of an integrable system.- Lax representation.- The Calogero system: inverse scattering.- Lie-Scheffers systems.- The inhomogeneous linear equation revisited.- Inhomogeneous linear systems.- Non-linear superposition rule.- Related maps.- Lie systems on Lie groups and homogeneous spaces.- Some examples of Lie systems.- Hamiltonian systems of Lie type.