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Geometry from Dynamics, Classical and Quantum

  • José F. Cariñena
  • Alberto Ibort
  • Giuseppe Marmo
  • Giuseppe Morandi

Table of contents

  1. Front Matter
    Pages i-xxv
  2. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 1-62
  3. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 63-134
  4. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 135-192
  5. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 193-270
  6. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 271-405
  7. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 407-487
  8. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 489-537
  9. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 539-568
  10. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 569-610
  11. José F. Cariñena, Alberto Ibort, Giuseppe Marmo, Giuseppe Morandi
    Pages 611-713
  12. Back Matter
    Pages 715-719

About this book

Introduction

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 superintegrability, 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.

Keywords

Integrable Systems Invariant Hermitean Structures Inverse Problems Kaehler Manifolds Kustaanheimo-Stiefel Fibration Lagrangian Structures Lax Representation Lie-Scheffers Systems Poisson Structures Symplectic Structures

Authors and affiliations

  • José F. Cariñena
    • 1
  • Alberto Ibort
    • 2
  • Giuseppe Marmo
    • 3
  • Giuseppe Morandi
    • 4
  1. 1.Departamento de Física TeóricaUniversidad de ZaragozaZaragozaSpain
  2. 2.Dept. MatemáticasUniversidad Carlos III de MadridMadridSpain
  3. 3.Departimento di Scienze FisicheUniversita Federico II di NapoliNapoliItaly
  4. 4.Universitá di Bologna INFN Sezione di BolognaBolognaItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-9220-2
  • Copyright Information Springer Science+Business Media Dordrecht 2015
  • Publisher Name Springer, Dordrecht
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-94-017-9219-6
  • Online ISBN 978-94-017-9220-2
  • Buy this book on publisher's site
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