Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems / Edition 2

Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems / Edition 2

by J.E. Marsden, Tudor Ratiu
     
 

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ISBN-10: 1441931430

ISBN-13: 9781441931436

Pub. Date: 12/01/2010

Publisher: Springer New York

Symmetry has always played an important role in mechanics, from fundamental formulations of basic principles to concrete applications. The theme of the book is to develop the basic theory and applications of mechanics with an emphasis on the role of symmetry. In recent times, the interest in mechanics, and in symmetry techniques in particular, has accelerated because

Overview

Symmetry has always played an important role in mechanics, from fundamental formulations of basic principles to concrete applications. The theme of the book is to develop the basic theory and applications of mechanics with an emphasis on the role of symmetry. In recent times, the interest in mechanics, and in symmetry techniques in particular, has accelerated because of developments in dynamical systems, the use of geometric methods and new applications to integrable and chaotic systems, control systems, stability and bifurcation, and the study of specific rigid, fluid, plasma and elastic systems. Introduction to Mechanics and Symmetry lays the basic foundation for these topics and includes numerous specific applications, making it beneficial to physicists and engineers. This text has specific examples and applications showing how the theory works, and up-to-date techniques, all of which makes it accessible to a wide variety of readers, expecially senior undergraduate and graduate students in mathematics, physics and engineering.

For this second edition, the text has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available on-line.

Product Details

ISBN-13:
9781441931436
Publisher:
Springer New York
Publication date:
12/01/2010
Series:
Texts in Applied Mathematics Series, #17
Edition description:
Softcover reprint of hardcover 2nd ed. 1999
Pages:
586
Product dimensions:
1.22(w) x 9.21(h) x 6.14(d)

Table of Contents

1 Introduction and Overview.- 1.1 Lagrangian and Hamiltonian Formalisms.- 1.2 The 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 Poincaré-Melnikov Method and Chaos.- 1.13 Resonances, Geometric Phases, and Control.- 2 Hamiltonian Systems on Linear Symplectic 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 Poincar–-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 Kähler 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-Poincar– 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-Poincar– 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.

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