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Classical Dynamics

Book Description

Recent advances in the study of dynamical systems have revolutionized the way that classical mechanics is taught and understood. Classical Dynamics: A Contemporary Approach is a new and comprehensive textbook that provides a complete description of this fundamental branch of physics. The authors cover all the material that one would expect to find in a standard graduate course: Lagrangian and Hamiltonian dynamics, canonical transformations, the Hamilton-Jacobi equation, perturbation methods, and rigid bodies. They also deal with more advanced topics such as the relativistic Kepler problem, Liouville and Darboux theorems, and inverse and chaotic scattering. A key feature of the book is the early introduction of geometric (differential manifold) ideas, as well as detailed treatment of topics in nonlinear dynamics (such as the KAM theorem) and continuum dynamics (including solitons). The book contains many worked examples and over 200 homework exercises. It will be an ideal textbook for graduate students of physics, applied mathematics, theoretical chemistry, and engineering, as well as a useful reference for researchers in these fields. A solutions manual is available exclusively for instructors.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedications
  6. Contents
  7. List of Worked Examples
  8. Preface
    1. Two Paths Through the Book
  9. 1. Fundamentals of Mechanics
    1. 1.1 Elementary Kinematics
      1. 1.1.1 Trajectories of Point Particles
      2. 1.1.2 Position, Velocity, and Acceleration
    2. 1.2 Principles of Dynamics
      1. 1.2.1 Newton’s Laws
      2. 1.2.2 The Two Principles
        1. Principle 1
        2. Principle 2
        3. Discussion
      3. 1.2.3 Consequences of Newton’s Equations
        1. Introduction
        2. Force is a Vector
    3. 1.3 One-Particle Dynamical Variables
      1. 1.3.1 Momentum
      2. 1.3.2 Angular Momentum
      3. 1.3.3 Energy and Work
        1. In Three Dimensions
        2. Application to One-Dimensional Motion
    4. 1.4 Many-Particle Systems
      1. 1.4.1 Momentum and Center of Mass
        1. Center of Mass
        2. Momentum
        3. Variable Mass
      2. 1.4.2 Energy
      3. 1.4.3 Angular Momentum
    5. 1.5 Examples
      1. 1.5.1 Velocity Phase Space and Phase Portraits
        1. The Cosine Potential
        2. The Kepler Problem
      2. 1.5.2 A System with Energy Loss
      3. 1.5.3 Noninertial Frames and the Equivalence Principle
        1. Equivalence Principle
        2. Rotating Frames
    6. Problems
  10. 2. Lagrangian Formulation of Mechanics
    1. 2.1 Constraints and Configuration Manifolds
      1. 2.1.1 Constraints
        1. Constraint Equations
        2. Constraints and Work
      2. 2.1.2 Generalized Coordinates
      3. 2.1.3 Examples of Configuration Manifolds
        1. The Finite Line
        2. The Circle
        3. The Plane
        4. The Two-Sphere (s[sup[2]])
        5. The Double Pendulum
        6. Discussion
    2. 2.2 Lagrange’s Equations
      1. 2.2.1 Derivation of Lagrange’s Equations
      2. 2.2.2 Transformations of Lagrangians
        1. Equivalent Lagrangians
        2. Coordinate Independence
        3. Hessian Condition
      3. 2.2.3 Conservation of Energy
      4. 2.2.4 Charged Particle in an Electromagnetic Field
        1. The Lagrangian
        2. A Time-Dependent Coordinate Transformation
    3. 2.3 Central Force Motion
      1. 2.3.1 The General Central Force Problem
        1. Statement of the Problem; Reduced Mass
        2. Reduction to Two Freedoms
        3. The Equivalent One-Dimensional Problem
      2. 2.3.2 The Kepler Problem
      3. 2.3.3 Bertrand’s Theorem
    4. 2.4 The Tangent Bundle TQ
      1. 2.4.1 Dynamics on TQ
        1. Velocities Do Not Lie in Q
        2. Tangent Spaces and the Tangent Bundle
        3. Lagrange’s Equations and Trajectories on TQ
      2. 2.4.2 TQ as a Differential Manifold
        1. Differential Manifolds
        2. Tangent Spaces and Tangent Bundles
        3. Application to Lagrange’s Equations
    5. Problems
  11. 3. Topics in Lagrangian Dynamics
    1. 3.1 The Variational Principle and Lagrange’s Equations
      1. 3.1.1 Derivation
        1. The Action
        2. Hamilton’s Principle
        3. Discussion
      2. 3.1.2 Inclusion of Constraints
    2. 3.2 Symmetry and Conservation
      1. 3.2.1 Cyclic Coordinates
        1. Invariant Submanifolds and Conservation of Momentum
        2. Transformations, Passive and Active
        3. Three Examples
      2. 3.2.2 Noether’s Theorem
        1. Point Transformations
        2. The Theorem
    3. 3.3 Nonpotential Forces
      1. 3.3.1 Dissipative Forces in the Lagrangian Formalism
        1. Rewriting the EL Equations
        2. The Dissipative and Rayleigh Functions
      2. 3.3.2 The Damped Harmonic Oscillator
      3. 3.3.3 Comment on Time-Dependent Forces
    4. 3.4 A Digression on Geometry
      1. 3.4.1 Some Geometry
        1. Vector Fields
        2. One-Forms
        3. The Lie Derivative
      2. 3.4.2 The Euler–Lagrange Equations
      3. 3.4.3 Noether’s Theorem
        1. One-Parameter Groups
        2. The Theorem
    5. Problems
  12. 4. Scattering and Linear Oscillations
    1. 4.1 Scattering
      1. 4.1.1 Scattering by Central Forces
        1. General Considerations
        2. The Rutherford Cross Section
      2. 4.1.2 The Inverse Scattering Problem
        1. General Treatment
        2. Example: Coulomb Scattering
      3. 4.1.3 Chaotic Scattering, Cantor Sets, and Fractal Dimension
        1. Two Disks
        2. Three Disks, Cantor Sets
        3. Fractal Dimension and Lyapunov Exponent
        4. Some Further Results
      4. 4.1.4 Scattering of a Charge by a Magnetic Dipole
        1. The Störmer Problem
        2. The Equatorial Limit
        3. The General Case
    2. 4.2 Linear Oscillations
      1. 4.2.1 Linear Approximation: Small Vibrations
        1. Linearization
        2. Normal Modes
      2. 4.2.2 Commensurate and Incommensurate Frequencies
        1. The Invariant Torus T
        2. The Poincaré Map
      3. 4.2.3 A Chain of Coupled Oscillators
        1. General Solution
        2. The Finite Chain
      4. 4.2.4 Forced and Damped Oscillators
        1. Forced Undamped Oscillator
        2. Forced Damped Oscillator
    3. Problems
  13. 5. Hamiltonian Formulation of Mechanics
    1. 5.1 Hamilton’s Canonical Equations
      1. 5.1.1 Local Considerations
        1. From the Lagrangian to the Hamiltonian
        2. A Brief Review of Special Relativity
        3. The Relativistic Kepler Problem
      2. 5.1.2 The Legendre Transform
      3. 5.1.3 Unified Coordinates on T*Q
        1. The ξ Notation
        2. Variational Derivation of Hamilton’s Equations
        3. Poisson Brackets
        4. Poisson Brackets and Hamiltonian Dynamics
    2. 5.2 Symplectic Geometry
      1. 5.2.1 The Cotangent Manifold
      2. 5.2.2 Two-Forms
      3. 5.2.3 The Symplectic Form ω
    3. 5.3 Canonical Transformations
      1. 5.3.1 Local Considerations
        1. Reduction on T*Q by Constants of the Motion
        2. Definition of Canonical Transformations
        3. Changes Induced by Canonical Transformations
        4. Two Examples
      2. 5.3.2 Intrinsic Approach
      3. 5.3.3 Generating Functions of Canonical Transformations
        1. Generating Functions
        2. The Generating Functions Gives the New Hamiltonian
        3. Generating Functions of Type
      4. 5.3.4 One-Parameter Groups of Canonical Transformations
        1. Infinitesimal Generators of One-Parameter Groups; Hamiltonian Flows
        2. The Hamiltonian Noether Theorem
        3. Flows and Poisson Brackets
    4. 5.4 Two Theorems: Liouville and Darboux
      1. 5.4.1 Liouville’s Volume Theorem
        1. Volume
        2. Integration on T*Q The Liouville Theorem
        3. Poincaré Invariants
        4. Density of States
      2. 5.4.2 Darboux’s Theorem
        1. The Theorem
        2. Reduction
    5. Problems
    6. Canonicity Implies PB Preservation
  14. 6. Topics in Hamiltonian Dynamics
    1. 6.1 The Hamilton–Jacobi Method
      1. 6.1.1 The Hamilton–Jacobi Equation
        1. Derivation
        2. Properties of Solutions
        3. Relation to the Action
      2. 6.1.2 Separation of Variables
        1. The Method of Separation
        2. Example: Charged Particle in a Magnetic Field
      3. 6.1.3 Geometry and the HJ Equation
      4. 6.1.4 The Analogy Between Optics and the HJ Method
    2. 6.2 Completely Integrable Systems
      1. 6.2.1 Action–Angle Variables
        1. Invariant Tori
        2. The ϕ and (J[sub[α]])
        3. The Canonical Transformation to AA Variables
        4. Example: A Particle on a Vertical Cylinder
      2. 6.2.2 Liouville’s Integrability Theorem
        1. Complete Integrability
        2. The Tori
        3. The (J[sub[α]])
        4. Example: the Neumann Problem
      3. 6.2.3 Motion on the Tori
        1. Rational and Irrational Winding Lines
        2. Fourier Series
    3. 6.3 Perturbation Theory
      1. 6.3.1 Example: The Quartic Oscillator; Secular Perturbation Theory
      2. 6.3.2 Hamiltonian Perturbation Theory
        1. Perturbation via Canonical Transformations
        2. Averaging
        3. Canonical Perturbation Theory in One Freedom
        4. Canonical Perturbation Theory in Many Freedoms
        5. The Lie Transformation Method
        6. Example: The Quartic Oscillator
    4. 6.4 Adiabatic Invariance
      1. 6.4.1 The Adiabatic Theorem
        1. Oscillator with Time-Dependent Frequency
        2. The Theorem
        3. Remarks on N > 1
      2. 6.4.2 Higher Approximations
      3. 6.4.3 The Hannay Angle
      4. 6.4.4 Motion of a Charged Particle in a Magnetic Field
        1. The Action Integral
        2. Three Magnetic Adiabatic Invariants
    5. Problems
  15. 7. Nonlinear Dynamics
    1. 7.1 Nonlinear Oscillators
      1. 7.1.1 A Model System
      2. 7.1.2 Driven Quartic Oscillator
        1. Damped Driven Quartic Oscillator; Harmonic Analysis
        2. Undamped Driven Quartic Oscillator
      3. 7.1.3 Example: The van der Pol Oscillator
    2. 7.2 Stability of Solutions
      1. 7.2.1 Stability of Autonomous Systems
        1. Definitions
        2. The Poincaré–Bendixon Theorem
        3. Linearization
      2. 7.2.2 Stability of Nonautonomous Systems
        1. The Poincaré Map
        2. Linearization of Discrete Maps
        3. Example: The Linearized Hénon Map
    3. 7.3 Parametric Oscillators
      1. 7.3.1 Floquet Theory
        1. The Floquet Operator R
        2. Standard Basis
        3. Eigenvalues of R and Stability
        4. Dependence on G
      2. 7.3.2 The Vertically Driven Pendulum
        1. The Mathieu Equation
        2. Stability of the Pendulum
        3. The Inverted Pendulum
        4. Damping
    4. 7.4 Discrete Maps; Chaos
      1. 7.4.1 The Logistic Map
        1. Definition
        2. Fixed Points
        3. Period Doubling
        4. Universality
        5. Further Remarks
      2. 7.4.2 The Circle Map
        1. The Damped Driven Pendulum
        2. The Standard Sine Circle Map
        3. Rotation Number and the Devil’s Staircase
        4. Fixed Points of the Circle Map
    5. 7.5 Chaos in Hamiltonian Systems and the KAM Theorem
      1. 7.5.1 The Kicked Rotator
        1. The Dynamical System
        2. The Standard Map
        3. Poincaré Map of the Perturbed System
      2. 7.5.2 The Hénon Map
      3. 7.5.3 Chaos in Hamiltonian Systems
        1. Poincaré–Birkhoff Theorem
        2. The Twist Map
        3. Numbers and Properties of the Fixed Points
        4. The Homoclinic Tangle
        5. The Transition to Chaos
      4. 7.5.4 The KAM Theorem
        1. Background
        2. Two Conditions: Hessian and Diophantine
        3. The Theorem
        4. A Brief Description of the Proof of KAM
    6. Problems
    7. Number Theory
      1. The Unit Interval
      2. A Diophantine Condition
      3. The Circle and the Plane
      4. KAM and Continued Fractions
  16. 8. Rigid Bodies
    1. 8.1 Introduction
      1. 8.1.1 Rigidity and Kinematics
        1. Definition
        2. The Angular Velocity Vector ω
      2. 8.1.2 Kinetic Energy and Angular Momentum
        1. Kinetic Energy
        2. Angular Momentum
      3. 8.1.3 Dynamics
        1. Space and Body Systems
        2. Dynamical Equations
        3. Example: The Gyrocompass
        4. Motion of the Angular Momentum J
        5. Fixed Points and Stability
        6. The Poinsot Construction
    2. 8.2 The Lagrangian and Hamiltonian Formulations
      1. 8.2.1 The Configuration Manifold (Q[sub[R]])
        1. Inertial, Space, and Body Systems
        2. The Dimension of (Q[sub[R]])
        3. The Structure of (Q[sub[R]])
      2. 8.2.2 The Lagrangian
        1. Kinetic Energy
        2. The Constraints
      3. 8.2.3 The Euler–Lagrange Equations
        1. Derivation
        2. The Angular Velocity Matrix Ω
      4. 8.2.4 The Hamiltonian Formalism
      5. 8.2.5 Equivalence to Euler’s Equations
        1. Antisymmetric Matrix–Vector Correspondence
        2. The Torque
        3. The Angular Velocity Pseudovector and Kinematics
        4. Transformations of Velocities
        5. Hamilton’s Canonical Equations
      6. 8.2.6 Discussion
    3. 8.3 Euler Angles and Spinning Tops
      1. 8.3.1 Euler Angles
        1. Definition
        2. R in Terms of the Euler Angles
        3. Angular Velocities
        4. Discussion
      2. 8.3.2 Geometric Phase for a Rigid Body
      3. 8.3.3 Spinning Tops
        1. The Lagrangian and Hamiltonian
        2. The Motion of the Top
        3. Nutation and Precession
        4. Quadratic Potential; the Neumann Problem
    4. 8.4 Cayley–Klein Parameters
      1. 8.4.1 2 × 2 Matrix Representation of 3-Vectors and Rotations
        1. 3-Vectors
        2. Rotations
      2. 8.4.2 The Pauli Matrices and CK Parameters
        1. Definitions
        2. Finding R[sub[U]])
        3. Axis and Angle in terms of the CK Parameters
      3. 8.4.3 Relation Between SU(2) and SO(3)
    5. Problems
  17. 9. Continuum Dynamics
    1. 9.1 Lagrangian Formulation of Continuum Dynamics
      1. 9.1.1 Passing to the Continuum Limit
        1. The Sine–Gordon Equation
        2. The Wave and Klein–Gordon Equations
      2. 9.1.2 The Variational Principle
        1. Introduction
        2. Variational Derivation of the EL Equations
        3. The Functional Derivative
        4. Discussion
      3. 9.1.3 Maxwell’s Equations
        1. Some Special Relativity
        2. Electromagnetic Fields
        3. The Lagrangian and the EL Equations
    2. 9.2 Noether’s Theorem and Relativistic Fields
      1. 9.2.1 Noether’s Theorem
        1. The Theorem
        2. Conserved Currents
        3. Energy and Momentum in the Field
        4. Example: The Electromagnetic Energy–Momentum Tensor
      2. 9.2.2 Relativistic Fields
        1. Lorentz Transformations
        2. Lorentz Invariant L and Conservation
        3. Free Klein–Gordon Fields
        4. Complex K–G Field and Interaction with the Maxwell Field
        5. Discussion of the Coupled Field Equations
      3. 9.2.3 Spinors
        1. Spinor Fields
        2. A Spinor Field Equation
    3. 9.3 The Hamiltonian Formalism
      1. 9.3.1 The Hamiltonian Formalism for Fields
        1. Definitions
        2. The Canonical Equations
        3. Poisson Brackets
      2. 9.3.2 Expansion in Orthonormal Functions
        1. Orthonormal Functions
        2. Particle-like Equations
        3. Example: Klein–Gordon
    4. 9.4 Nonlinear Field Theory
      1. 9.4.1 The Sine–Gordon Equation
        1. Soliton Solutions
        2. Properties of sG Solitons
        3. Multiple-Soliton Solutions
        4. Generating Soliton Solutions
        5. Nonsoliton Solutions
        6. Josephson Junctions
      2. 9.4.2 The Nonlinear K–G Equation
        1. The Lagrangian and the EL Equation
        2. Kinks
    5. 9.5 Fluid Dynamics
      1. 9.5.1 The Euler and Navier–Stokes Equations
        1. Substantial Derivative and Mass Conservation
        2. Euler’s Equation
        3. Viscosity and Incompressibility
        4. The Navier–Stokes Equations
        5. Turbulence
      2. 9.5.2 The Burgers Equation
        1. The Equation
        2. Asymptotic Solution
      3. 9.5.3 Surface Waves
        1. Equations for the Waves
        2. Linear Gravity Waves
        3. Nonlinear Shallow Water Waves: the KdV Equation
        4. Single KdV Solitons
        5. Multiple KdV Solitons
    6. 9.6 Hamiltonian Formalism for Nonlinear Field Theory
      1. 9.6.1 The Field Theory Analog of Particle Dynamics
        1. From Particles to Fields
        2. Dynamical Variables and Equations of Motion
      2. 9.6.2 The Hamiltonian Formalism
        1. The Gradient
        2. The Symplectic Form
        3. The Condition for Canonicity
        4. Poisson Brackets
      3. 9.6.3 The kdV Equation
        1. KdV as a Hamiltonian Field
        2. Constants of the Motion
        3. Generating the Constants of the Motion
        4. More on Constants of the Motion
      4. 9.6.4 The Sine–Gordon Equation
        1. Two-Component Field Variables
        2. sG as a Hamiltonian Field
    7. Problems
  18. Epilogue
  19. Appendix: Vector Spaces
    1. General Vector Spaces
    2. Linear Operators
    3. Inverses and Eigenvalues
    4. Inner Products and Hermitian Operators
  20. Bibliography
  21. Index