Marsden J.E., Ratiu T.S. Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems

Califoia, 1998. 535 p.
Contents
Preface
About the Authors
The Book
Introduction and Overview
Lagrangian and Hamiltonian Formalisms
The Rigid Body
Lie Poisson Brackets, Poisson Manifolds, Momentum Maps
The Heavy Top
Incompressible Fluids
The Maxwell Vlasov System
Nonlinear Stability
Bifurcation
The Poincar¶ e{Melnikov Method
Resonances, Geometric Phases, and Control
Hamiltonian Systems on Linear Symplectic Spaces
Introduction
Symplectic Forms on Vector Spaces
Canonical Transformations or Symplectic Maps
The General Hamilton Equations
When Are Equations Hamiltonian?
Hamiltonian Flows
Poisson Brackets
A Particle in a Rotating Hoop
The Poincare Melnikov Method and Chaos
An Introduction to In?nite-Dimensional Systems
Lagrange's and Hamilton's Equations for Field Theory
Examples: Hamilton's Equations .
Examples: Poisson Brackets and Conserved Quantities. . .
Interlude: Manifolds, Vector Fields, and Diferential Forms
Manifolds
Diferential Forms
The Lie Derivative
Stokes' Theorem Hamiltonian Systems on Symplectic Manifolds
Symplectic Manifolds
Symplectic Transformations
Complex Structures and K? ahler Manifolds
Hamiltonian Systems
Poisson Brackets on Symplectic Manifolds Cotangent Bundles
The Linear Case
The Nonlinear Case
Cotangent Lifts
Lifts of Actions
Generating Functions
Fiber Translations and Magnetic Terms
A Particle in a Magnetic Field
Lagrangian Mechanics
Hamilton's Principle of Critical Action
The Legendre Transform
Euler Lagrange Equations
Hyperregular Lagrangians and Hamiltonians
Geodesics
The Kaluza{Klein Approach to Charged Particles
Motion in a Potential Field
The Lagrange d'Alembert Principle
The Hamilton{Jacobi Equation
Variational Principles, Constraints, and Rotating Systems
A Retu to Variational Principles
The Geometry of Variational Principles
Constrained Systems
Constrained Motion in a Potential Field
Dirac Constraints
Centrifugal and Coriolis Forces
The Geometric Phase for a Particle in a Hoop
Moving Systems
Routh Reduction
An Introduction to Lie Groups
Basic De?nitions and Properties
Some Classical Lie Groups
Actions of Lie Groups
Poisson Manifolds
The De?nition of Poisson Manifolds
Hamiltonian Vector Fields and Casimir Functions
Properties of Hamiltonian Flows
The Poisson Tensor
Quotients of Poisson Manifolds
The Schouten Bracket
Generalities on Lie Poisson Structures
Momentum Maps
Canonical Actions and Their Initesimal Generators
Momentum Maps
An Algebraic Denition of the Momentum Map
Conservation of Momentum Maps
Equivariance of Momentum Maps
Computation and Properties of Momentum Maps
Momentum Maps on Cotangent Bundles
Examples of Momentum Maps
Equivariance and In?nitesimal Equivariance
Equivariant Momentum Maps Are Poisson
Poisson Automorphisms
Momentum Maps and Casimir Functions
Lie Poisson and Euler Poincare Reduction
The Lie Poisson Reduction Theorem
Proof of the Lie{Poisson Reduction Theorem for GL(n)
Proof of the Lie{Poisson Reduction Theorem for Divol
Lie Poisson Reduction using Momentum Functions
Reduction and Reconstruction of Dynamics
The Linearized Lie Poisson Equations
The Euler Poincare Equations
The Lagrange Poincare Equations
Coadjoint Orbits
Examples of Coadjoint Orbits
Tangent Vectors to Coadjoint Orbits
The Symplectic Structure on Coadjoint Orbits
The Orbit Bracket via Restriction of the Lie Poisson Bracket
The Special Linear Group on the Plane
The Euclidean Group of the Plane
The Euclidean Group of Three-Space
The Free Rigid Body
Material, Spatial, and Body Coordinates
The Lagrangian of the Free Rigid Body
The Lagrangian and Hamiltonian in Body Representation
Kinematics on Lie Groups
Poinsots Theorem
Euler Angles
The Hamiltonian of the Free Rigid Body in the Material
Description via Euler Angles
The Analytical Solution of the Free Rigid Body Problem
Rigid Body Stability
Heavy Top Stability
The Rigid Body and the Pendulum