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| 《INTRODUCTILN TOMECHAHICS AND SYMMETRY》 | |
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内容简介
Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modem as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses.
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目录
目 录 Preface 1 Introduction and Overview 1.1 Lagrangian and Hamiltonian Formalisms 1.2 Tile Rigid Body 1.3 Lie-Poisson Brackets, Poisson Manifolds, Momentum Maps 1.4 Incompressible Fluids 1.5 The Maxwell-Vlasov System 1.6 The Maxwell and Poisson-Vlasov Brackets 1.7 The Poisson-Vlasov to Fluid Map 1.8 The Maxwell-Vlasov Bracket 1.9 The Heavy Top 1.10 Nonlinear Stability 1.11 Bifurcation 1.12 The Poincare-Melnikov Method and Chaos 1.13 Resonances, Geometric Phases, and Control 2 Hamiltonian Systems on Linear Syrnplectic Spaces 2.1 Introduction 2.2 Symplectic Forms on Vector Spaces 2.3 Examples 2.4 Canonical Transformations or Symplectic Maps 2.5 The Abstract Hamilton Equations 2.6 The Classical Hamilton Equations 2.7 When Are Equations Hamiltonian 2.8 Hamiltonian Flows 2.9 Poisson Brackets 2.10 A Particle in a Rotating Hoop 2.11 The Poincare-Melnikov Method and Chaos 3 An Introduction to Infinite-Dimensional Systems 3.1 Lagrange''''s and Hamilton''''s Equations for Field Theory 3.2 Examples: Hamilton''''s Equations 3.3 Examples: Poisson Brackets and Conserved Quantities 4 Interlude: Manifolds, Vector Fields, Differential Forms 4.1 Manifolds 4.2 Differential Forms 4.3 The Lie Derivative 4.4 Stokes'''' Theorem 5 Hamiltonian Systems on Symplectic Manifolds 5.1 Symplectic Manifolds 5.2 Symplectic Transformations 5.3 Complex Structures and Kahler Manifolds 5.4 Hamiltonian Systems 5.5 Poisson Brackets on Symplectic Manifolds 6 Cotangent Bundles 6.1 The Linear Case 6.2 The Nonlinear Case 6.3 Cotangent Lifts 6.4 Lifts of Actions 6.5 Generating Functions 6.6 Fiber Translations and Magnetic Terms 6.7 A Particle in a Magnetic Field 6.8 Linearization of Hamiltonian Systems 7 Lagrangian Mechanics 7.1 The Principle of Critical Action 7.2 The Legendre Transform 7.3 Lagrange''''s Equations 7.4 Hyperregular Lagrangians and Hamiltonians 7.5 Geodesics 7.6 The Kaluza-Klein Approach to Charged Particles 7.7 Motion in a Potential Field 7.8 The Lagrange-d''''Alembert Principle 7.9 The Hamilton-Jacobi Equation 7.10 The Classical Limit and the Maslov Index 8 Variational Principles, Constraints, Rotating Systems 8.1 A Return to Variational Principles 8.2 The Lagrange Multiplier Theorem 8.3 Holonomic Constraints 8.4 Constrained Motion in a Potential Field 8.5 Dirac Constraints 8.6 Centrifugal and Coriolis Forces 8.7 The Geometric Phase for a Particle in a Hoop 8.8 The General Theory of Moving Systems 9 An Introduction to Lie Groups 9.1 Basic Definitions and Properties 9.2 Some Classical Lie Groups 9.3 Actions of Lie Groups 10 Poisson Manifolds 10.1 The Definition of Poisson Manifolds 10.2 Examples 10.3 Hamiltonian Vector Fields and Casimir Functions 10.4 Examples 10.5 Properties of''''Hamiltonian Flows 10.6 The Poisson Tensor 10.7 Quotients of Poisson Manifolds 10.8 The Schouten Bracket 10.9 Generalities on Lie-Poisson Structures 11 Momentum Maps 11.1 Canonical Actions and Their Infinitesimal Generators 11.2 Momentum Maps 11.3 An Algebraic Definition of the Momentum Map 11.4 Conservation of Momentum Maps 11.5 Examples 11.6 Equivariance of Momentum Maps 12 Computation and Properties of Momentum Maps 12.1 Momentum Maps on Cotangent Bundles 12.2 Momentum Maps on Tangent Bundles 12.3 Examples 12.4 Equivariance and Infinitesimal Equivariance 12.5 Equivariant Momentum Maps Are Poisson 12.6 More Examples 12.7 Poisson Automorphisms 12.8 Momentum Maps and Casimir Functions 13 Euler-Poincare and Lie-Poisson Reduction 13. 1 The Lie-Poisson Reduction Theorem 13.2 Proof of the Lie-Poisson Reduction Theorem for GL n 13.3 Proof of the Lie-Poisson Reduction Theorem for Diffvol M 13.4 Proof of the Lie-Poisson Reduction Theorem for Diffcan P 13.5 Lie-Poisson Reduction Using Momentum Functions 13.6 Reduction and Reconstruction of Dynamics 13.7 The Linearized Lie-Poisson Bracket 13.8 The Euler-Poincare Equations 13.9 The Reduced Euler-Lagrange Equations 14 Coadjoint Orbits 14.1 Examples of Coadjoint Orbits 14.2 Tangent Vectors to Coadjoint Orbits 14.3 Examples of Tangent Vectors 14.4 The Symplectic Structure on Coadjoint Orbits 14.5 Examples of Symplectic Structures on Orbits 14.6 The Orbit Bracket via Restriction 14.7 The Special Linear Group on the Plane 14.8 The Euclidean Group of the Plane 14.9 The Euclidean Group of Three-Space 15 The Free Rigid Body 15.1 Material, Spatial, and Body Coordinates 15.2 The Lagrangian of the Free Rigid Body 15.3 The Lagrangian and Hamiltonian for the Rigid Body in Body Representation 15.4 Kinematics on Lie Groups 15.5 Poinsot''''s Theorem 15.6 Euler Angles 15.7 The Hamiltonian of the Free Rigid Body in the Material DeScription via Euler Angles 15.8 The Analytical Solution of the Free Rigid Body Problem 15.9 Rigid Body Stability 15.10 Heavy Top Stability 15.11 The Rigid Body and the Pendulum References Index
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