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Dordrecht : Springer, [2015]

xxv, 719 stran : ilustrace ; 25 cm

ISBN 978-94-017-9219-6 (vázáno)

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001418006

José F. Carinena ¦ Alberto lbort ¦ Giuseppe Marmo • Giuseppe Morandi // Geometry from Dynamics, Classical and Quantum // 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. // Physics // ISBN 978-94-017-9219-6 // Contents // 1 Some Examples of Linear and Nonlinear Physical Systems // and Their Dynamical Equations... 1 // 1.1 Introduction... 1 // ? 1.2 Equations of Motion for Evolution Systems... 2 // 1.2.1 Histories, Evolution and Differential Equations... 2 // 1.2.2 The Isotropic Harmonic Oscillator... 4 // 1.2.3 Inhomogeneous or Affine Equations... 5 // 1.2.4 A Free Falling Body in a Constant Force Field... 7 // 1.2.5 Charged Particles in Uniform and Stationary Electric // and Magnetic Fields... 8 // 1.2.6 Symmetries and Constants of Motion... 12 // 1.2.7 The Non-isotropic Harmonic Oscillator... 16 // 1.2.8 Lagrangian and Hamiltonian Descriptions // of Evolution Equations... 21 // 1.2.9 The Lagrangian Descriptions of the Harmonic // Oscillator... 27 // 1.2.10 Constructing Nonlinear Systems Out of

Linear Ones ... 28 // 1.2.11 The Reparametrized Harmonic Oscillator... 29 // 1.2.12 Reduction of Linear Systems... 34 // Ł 1.3 Linear Systems with Infinite Degrees of Freedom... 41 // 1.3.1 The Klein-Gordon Equation and the Wave Equation ... 41 // 1.3.2 The Maxwell Equations... 44 // 1.3.3 The Schrödinger Equation... 50 // 1.3.4 Symmetries and Infinite-Dimensional Systems... 53 // 1.3.5 Constants of Motion... 55 // ¦ References... 61 // 2 The Language of Geometry and Dynamical Systems: // The Linearity Paradigm... 63 // 2.1 Introduction... 63 // 2.2 Linear Dynamical Systems: The Algebraic Viewpoint... 64 // XIX // XX // Contents // 2.2.1 Linear Systems and Linear Spaces... 64 // 2.2.2 Integrating Linear Systems: Linear Flows... 66 // 2.2.3 Linear Systems and Complex Vector Spaces... 73 // 2.2.4 Integrating Time-Dependent Linear Systems: // Dyson’s Formula... 79 // 2.2.5 From a Vector Space to Its Dual: Induced Evolution // Equations... 82 // 2.3 From Linear Dynamical Systems to Vector Fields... 84 // 2.3.1 Flows in the Algebra of Smooth Functions... 84 // 2.3.2 Transformations and Flows... 86 // 2.3.3 The Dual Point of View of Dynamical Evolution... 87 // 2.3.4 Differentials and Vector Fields: Locality... 89 // 2.3.5 Vector Fields and Derivations on the Algebra // of Smooth Functions... 91 // 2.3.6 The ‘Heisenberg’ Representation of Evolution... 93 // 2.3.7 The Integration Problem for Vector Fields... 95 // 2.4 Exterior Differential Calculus on Linear Spaces... 100

2.4.1 Differential Forms... 100 // 2.4.2 Exterior Differential Calculus: Cartan Calculus... 102 // 2.4.3 The ‘Easy’ Tensorialization Principle... 108 // 2.4.4 Closed and Exact Forms... Ill // 2.5 The General ‘Integration’ Problem for Vector Fields... 113 // 2.5.1 The Integration Problem for Vector Fields: // Frobenius Theorem... 113 // 2.5.2 Foliations and Distributions... 115 // 2.6 The Integration Problem for Lie Algebras... 118 // 2.6.1 Introduction to the Theory of Lie Groups: // Matrix Lie Groups... 119 // 2.6.2 The Integration Problem for Lie Algebras*... 130 // References... 134 // 3 The Geometrization of Dynamical Systems... 135 // 3.1 Introduction... 135 // 3.2 Differentiable Spaces*... 137 // 3.2.1 Ideals and Subsets... 138 // 3.2.2 Algebras of Functions and Differentiable Algebras... 141 // 3.2.3 Generating Sets... 143 // 3.2.4 Infinitesimal Symmetries and Constants of Motion... 145 // 3.2.5 Actions of Lie Groups and Cohomology... 147 // 3.3 The Tensorial Characterization of Linear Structures // and Vector Bundles... 153 // 3.3.1 A Tensorial Characterization of Linear Structures... 153 // 3.3.2 Partial Linear Structures... 157 // 3.3.3 Vector Bundles... 159 // Bntents xxi // I 3.4 The Holonomic Tensorialization Principle*... 163 // 3.4.1 The Natural Tensorialization of Algebraic Structures ... 163 // 3.4.2 The Holonomic Tensorialization Principle... 165 // 3.4.3 Geometric Structures Associated to Algebras... 169 // ¦ 3.5 Vector Fields and Linear Structures...

171 // 3.5.1 Linearity and Evolution... 171 // 3.5.2 Linearizable Vector Fields... 172 // 3.5.3 Alternative Linear Structures: Some Examples... 175 // ¦ 3.6 Normal Forms and Symmetries... 180 // 3.6.1 The Conjugacy Problem... 180 // 3.6.2 Separation of Vector Fields... 184 // 3.6.3 Symmetries for Linear Vector Fields... 186 // 3.6.4 Constants of Motion for Linear Dynamical Systems... 188 // 4 References... 192 // 4 Invariant Structures for Dynamical Systems: Poisson Dynamics. . . 193 // ¦ 4.1 Introduction... 193 // I 4.2 The Factorization Problem for Vector Fields... 194 // 4.2.1 The Geometry of Noether’s Theorem... 194 // 4.2.2 Invariant 2-Tensors... 195 // 4.2.3 Factorizing Linear Dynamics: Linear Poisson // Factorization... 200 // I 4.3 Poisson Structures... 210 // 4.3.1 Poisson Algebras and Poisson Tensors... 210 // 4.3.2 The Canonical ‘Distribution’ of a Poisson Structure... 214 // 4.3.3 Poisson Structures and Lie Algebras... 215 // 4.3.4 The Coadjoint Action and Coadjoint Orbits... 219 // 4.3.5 The Heisenberg-Weyl, Rotation and Euclidean // Groups... 221 // 14.4 Hamiltonian Systems and Poisson Structures... 227 // 4.4.1 Poisson Tensors Invariant Under Linear Dynamics... 227 // 4.4.2 Poisson Maps... 231 // 4.4.3 Symmetries and Constants of Motion... 233 // ? 4.5 The Inverse Problem for Poisson Structures: // Feynman’s Problem... 243 // 4.5.1 Alternative Poisson Descriptions... 244 // 4.5.2 Feynman’s Problem... 247 // 4.5.3 Poisson Description of Internal Dynamics...

249 // 4.5.4 Poisson Structures for Internal and External Dynamics. . . 253 // ¦ 4.6 The Poincaré Group and Massless Systems... 260 // 4.6.1 The Poincaré Group... 260 // 4.6.2 A Classical Description for Free Massless Particles... 267 // I References... 269 // xxii Conienis // 5 The Classical Formulations of Dynamics of Hamilton // and Lagrange... 271 // 5.1 Introduction... 271 // 5.2 Linear Hamiltonian Systems... 272 // 5.2.1 Symplectic Linear Spaces... 273 // 5.2.2 The Geometry of Symplectic Linear Spaces... 276 // 5.2.3 Generic Subspaces of Symplectic Linear Spaces... 281 // 5.2.4 Transformations on a Symplectic Linear Space... 282 // 5.2.5 On the Stmcture of the Group Sp(«)... 286 // 5.2.6 Invariant Symplectic Structures... 288 // 5.2.7 Normal Forms for Hamiltonian Linear Systems... 292 // 5.3 Symplectic Manifolds and Hamiltonian Systems... 295 // 5.3.1 Symplectic Manifolds... 295 // 5.3.2 Symplectic Potentials and Vector Bundles... 300 // 5.3.3 Hamiltonian Systems of Mechanical Type... 303 // 5.4 Symmetries and Constants of Motion // for Hamiltonian Systems... 305 // 5.4.1 Symmetries and Constants of Motion: // The Linear Case... 305 // 5.4.2 Symplectic Realizations of Poisson Structures... 306 // 5.4.3 Dual Pairs and the Cotangent Group... 308 // 5.4.4 An Illustrative Example: The Harmonic Oscillator... 311 // 5.4.5 The 2-Dimensional Harmonic Oscillator... 312 // 5.5 Lagrangian Systems... 320 // 5.5.1 Second-Order Vector Fields... 321 // 5.5.2 The Geometry of the Tangent

Bundle... 326 // 5.5.3 Lagrangian Dynamics... 341 // 5.5.4 Symmetries. Constants of Motion // and the Noether Theorem... 351 // 5.5.5 A Relativistic Description for Massless Particles... 358 // 5.6 Feynman’s Problem and the Inverse Problem // for Lagrangian Systems... 360 // 5.6.1 Feynman’s Problem Revisited... 360 // 5.6.2 Poisson Dynamics on Bundles and the Inclusion // of Internal Variables... 366 // 5.6.3 The Inverse Problem for Lagrangian Dynamics... 374 // 5.6.4 Feynman’s Problem and Lie Groups... 383 // References... 404 // 6 The Geometry of Hermitean Spaces: Quantum Evolution... 407 // 6.1 Summary... 407 // 6.2 Introduction... 407 // xxiii // 6.3 Invariant Hermitean Structures... 409 // í 6.3.1 Positive-Factorizable Dynamics... 409 // I 6.3.2 Invariant Hermitean Metrics... 417 // 6.3.3 Hennitean Dynamics and Its Stability Properties... 420 // 6.3.4 Bihamiltonian Descriptions... 421 // 6.3.5 The Structure of Compatible Hermitean Forms... 424 // 6.4 Complex Structures and Complex Exterior Calculus... 430 // 6.4.1 The Ring of Functions of a Complex Space... 430 // 6.4.2 Complex Linear Systems... 433 // 6.4.3 Complex Differential Calculus and Kähler Manifolds. . . . 435 // 6.4.4 Algebras Associated with Hermitean Structures... 437 // 6.5 The Geometry of Quantum Dynamical Evolution... 439 // 6.5.1 On the Meaning of Quantum Dynamical Evolution... 439 // 6.5.2 The Basic Geometry of the Space of Quantum States . . . 444 // 6.5.3 The Hermitean Structure on the Space of

Rays... 448 // 6.5.4 Canonical Tensors on a Hilbert Space... 449 // 6.5.5 The Kähler Geometry of the Space of Pure // Quantum States... 453 // 6.5.6 The Momentum Map and the Jordan-Scwhinger Map . . . 456 // 6.5.7 A Simple Example: The Geometry of a Two-Level // System... 459 // I 6.6 The Geometry of Quantum Mechanics and the GNS // Construction... 462 // 6.6.1 The Space of Density States... 463 // 6.6.2 The GNS Construction... 467 // 6.7 Alternative Hermitean Structures for Quantum Systems... 471 // 6.7.1 Equations of Motion on Density States // and Hermitean Operators... 471 // 6.7.2 The Inverse Problem in Various Formalisms... 471 // 6.7.3 Alternative Hermitean Structures for Quantum Systems: // The Infinite-Dimensional Case... 481 // Bteferences... 485 // 7 Folding and Unfolding Classical and Quantum Systems... 489 // 7.1 Introduction... 489 // 7.2 Relationships Between Linear and Nonlinear Dynamics... 489 // 7.2.1 Separable Dynamics... 490 // 7.2.2 The Riccati Equation... 491 // 7.2.3 Burgers Equation... 493 // 7.2.4 Reducing the Free System Again... 495 // 7.2.5 Reduction and Solutions of the Hamilton-Jacobi // Equation... 499 // XXIV // Contents // 7.3 The Geometrical Description of Reduction... 500 // 7.3.1 A Charged Non-relativistic Particle in a Magnetic // Monopole Field... 503 // 7.4 The Algebraic Description... 504 // 7.4.1 Additional Structures: Poisson Reduction... 506 // 7.4.2 Reparametrization of Linear Systems... 508 // 7.4.3 Regularization and Linearization

of the Kepler // Problem... 514 // 7.5 Reduction in Quantum Mechanics... 520 // 7.5.1 The Reduction of Free Motion in the Quantum Case ... 520 // 7.5.2 Reduction in Terms of Differential Operators... 522 // 7.5.3 The Kustaanheimo-Stiefel Fibration... 524 // 7.5.4 Reduction in the Heisenberg Picture... 527 // 7.5.5 Reduction in the Ehrenfest Formalism... 532 // References... 535 // 8 Integrable and Superintegrable Systems... 539 // 8.1 Introduction: What Is Integrability?... 539 // 8.2 A First Approach to the Notion of Integrability: Systems // with Bounded Trajectories... 541 // 8.2.1 Systems with Bounded Trajectories... 542 // 8.3 The Geometrization of the Notion of Integrability... 546 // 8.3.1 The Geometrical Notion of Integrability // and the Erlangen Programme... 548 // 8.4 A Normal Form for an Integrable System... 550 // 8.4.1 Integrability and Alternative Hamiltonian Descriptions.. . 550 // 8.4.2 Integrability and Normal Forms... 552 // 8.4.3 The Group of Diffeomorphisms of an Integrable // System... 555 // 8.4.4 Oscillators and Nonlinear Oscillators... 556 // 8.4.5 Obstructions to the Equivalence of Integrable Systems. . . 557 // 8.5 Lax Representation... 558 // 8.5.1 The Toda Model... 561 // 8.6 The Calogero System: Inverse Scattering... 563 // 8.6.1 The Integrability of the Calogero-Moser System... 563 // 8.6.2 Inverse Scattering: A Simple Example... 564 // 8.6.3 Scattering States for the Calogero System... 565 // References... 567 // 9 Lie-SchefFers Systems... 569

9.1 The Inhomogeneous Linear Equation Revisited... 569 // 9.2 Inhomogeneous Linear Systems... 571 // 9.3 Non-linear Superposition Rule... 578 // 9.4 Related Maps... 581 // XXV // ents // .5 Lie-Scheffers Systems on Lie Groups and Homogeneous // Spaces... 583 // .6 Some Examples of Lie-Scheffers Systems... 589 // 9.6.1 Riccati Equation... 589 // 9.6.2 Euler Equations... 595 // 9.6.3 SODE Lie-Scheffers Systems... 597 // 9.6.4 Schrödinger-Pauli Equation... 598 // 9.6.5 Smorodinsky-Wintemitz Oscillator... 599 // 9.7 Hamiltonian Systems of Lie-Scheffers Type... 600 // 9.8 A Generalization of Lie-Scheffers Systems... 605 // eferences... 608 // 10 Appendices... 611 // ¦ References... 712 // Index // 715