1.Elementary Newtonian Mechanics
1.1 Newton''s Laws 1687 and Their Interpretation
1.2 Uniform Rectilinear Motion and Inertial Systems
1.3 Inertial Frames in Relative Motion
1.4 Momentum and Force
1.5 Typical Forces. A Remark About Units
1.6 Space, Time, and Forces
1.7 The Two-Body System with Internal Forces
1.7.1 Center-of-Mass and Relative Motion
1.7.2 Example: The Gravitational Force Between Two Celestial Bodies Kepler''s Problem
1.7.3 Center-of-Mass and Relative Momentum in the Two-Body System
1.8 Systems of Finitely Many Particles
1.9 The Principle of Center-of-Mass Motion
1.10 The Principle of Angular-Momentum Conservation
1.11 The Principle of Energy Conservation
1.12 The Closed n-Particle System
1.13 Galilei Transformations
1.14 Space and Time with Galilei Invariance
1.15 Conservative Force Fields
1.16 One-Dimensional Motion of a Point Particle
1.17 Examples of Motion in One Dimension
1.17.1 The Harmonic Oscillator
1.17.2 The Planar Mathematical Pendulum
1.18 Phase Space for the n-Particle System in R3
1.19 Existence and Uniqueness of the Solutions of x_" = ~_x, t
1.20 Physical Consequences of the Existence and Uniqueness Theorem
1.21 Linear Systems
1.21.1 Linear, Homogeneous Systems
1.21.2 Linear, Inhomogeneous Systems
1.22 Integrating One-Dimensional Equations of Motion
1.23 Example: The Planar Pendulum for Arbitrary Deviations from the Vertical
1.24 Example: The Two-Body System with a Central Force
1.25 Rotating Reference Systems: Coriolis and Centrifugal Forces
1.26 Examples of Rotating Reference Systems
1.27 Scattering of Two Particles that Interact via a Central Force: Kinematics
1.28 Two-Particle Scattering with a Central Force: Dynamics
1.29 Example: Coulomb Scattering of Two Particles with Equal Mass and Charge
1.30 Mechanical Bodies of Finite Extension
1.31 Time Averages and the Virial Theorem Appendix: Practical Examples
2. The Principles of Canonical Mechanics
2.1 Constraints and Generalized Coordinates
2.1.1 Definition of Constraints
2.1.2 Generalized Coordinates
2.2 D''Alembert''s Principle
2.2.1 Definition of Virtual Displacements
2.2.2 The Static Case
2.2.3 The Dynamical Case
2.3 Lagrange''s Equations
2.4 Examples of the Use of Lagrange''s Equations
2.5 A Digression on Variational Principles
2.6 Hamilton''s Variational Principle 1834
2.7 The Euler-Lagrange Equations
2.8 Further Examples of the Use of Lagrange''s Equations
2.9 A Remark About Nonuniqueness of the Lagrangian Function
2.10 Gauge Transformations of the Lagrangian Function
2.11 Admissible Transformations of the Generalized Coordinates
2.12 The Hamiltonian Function and Its Relation to the Lagrangian Function L
2.13 The Legendre Transformation for the Case of One Variable
2.14 The Legendre Transformation for the Case of Several Variables
2.15 Canonical Systems
2.16 Examples of Canonical Systems
2.17 The Variational Principle Applied to the Hamiltonian Function
2.18 Symmetries and Conservation Laws
2.19 Noether''s Theorem
2.20 The Generator for Infinitesimal Rotations About an Axis
2.21 More About the Rotation Group
2.22 Infinitesimal Rotations and Their Generators
2.23 Canonical Transformations
2.24 Examples of Canonical Transformations
2.25 The Structure of the Canonical Equations
2.26 Example: Linear Autonomous Systems in One Dimension
2.27 Canonical Transformations in Compact Notation
2.28 On the Symplectic Structure of Phase Space
2.29 Liouville''s Theorem
2.29.1 The Local Form
2.29.2 The Global Form
2.30 Examples for the Use of Liouville''s Theorem
2.31 Poisson Brackets
2.32 Properties of Poisson Brackets
2.33 Infinitesimal Canonical Transformations
2.34 Integrals of the Motion
2.35 The Hamilton-Jacobi Differential Equation
2.36 Examples for the Use of the Hamilton-Jacobi Equation
2.37 The Hamilton-Jacobi Equation and Integrable Systems
2.37.1 Local Rectification of Hamiltonian Systems
2.37.2 Integrable Systems
2.37.3 Angle and Action Variables
2.38 Perturbing Quasiperiodic Hamiltonian Systems
2.39 Autonomous, Nondegenerate Hamiltonian Systems in the Neighborhood of Integrable Systems
2.40 Examples. The Averaging Principle
2.40.1 The Anharmonic Oscillator
2.40.2 Averaging of Perturbations
2.41 Generalized Theorem of Noether Appendix: Practical Examples
3. The Mechanics of Rigid Bodies
3.1 Definition of Rigid Body
3.2 Infinitesimal Displacement of a Rigid Body
3.3 Kinetic Energy and the Inertia Tensor
3.4 Properties of the Inertia Tensor
3.5 Steiner''s Theorem
3.6 Examples of the Use of Steiner''s Theorem
3.7 Angular Momentum of a Rigid Body
3.8 Force-Free Motion of Rigid Bodies
3.9 Another Parametrization of Rotations: The Euler Angles
3.10 Definition of Eulerian Angles
3.11 Equations of Motion of Rigid Bodies
3.12 Euler''s Equations of Motion
3.13 Euler''s Equations Applied to a Force-Free Top
3.14 The Motion of a Free Top and Geometric Constructions
3.15 The Rigid Body in the Framework of Canonical Mechanics
3.16 Example: The Symmetric Children''s Top in a Gravitational Field
3.17 More About the Spinning Top
3.18 Spherical Top with Friction: The ''Tippe Top".
3.18.1 Conservation Law and Energy Considerations
3.18.2 Equations of Motion and Solutions with Constant Energy
Appendix: Practical Examples
4. Relativistic Mechanics
4.1 Failures of Nonrelativistic Mechanics
4.2 Constancy of the Speed of Light
4.3 The Lorentz Transformations
4.4 Analysis of Lorentz and Poincar6 Transformations
4.4.1 Rotations and Special Lorentz Tranformations "Boosts"
4.4.2 Interpretation of Special Lorentz Transformations
4.5 Decomposition of Lorentz Transformations
into Their Components
4.5.1 Proposition on Orthochronous, Proper Lorentz Transformations
4.5.2 Corollary of the Decomposition Theorem and Some Consequences
4.6 Addition of Relativistic Velocities
4.7 Galilean and Lorentzian Space-Time Manifolds
4.8 Orbital Curves and Proper Time
4.9 Relativistic Dynamics
4.9.1 Newton''s Equation
4.9.2 The Energy-Momentum Vector
4.9.3 The Lorentz Force
4.10 Time Dilatation and Scale Contraction
4.11 More About the Motion of Free Particles
4.12 The Conformal Group
5. Geometric Aspects of Mechanics
5.1 Manifolds of Generalized Coordinates
5.2 Differentiable Manifolds
5.2.1 The Euclidean Space Rn
5.2.2 Smooth or Differentiable Manifolds
5.2.3 Examples of Smooth Manifolds
5.3 Geometrical Objects on Manifolds
5.3.1 Functions and Curves on Manifolds
5.3.2 Tangent Vectors on a Smooth Manifold
5.3.3 The Tangent Bundle of a Manifold
5.3.4 Vector Fields on Smooth Manifolds
5.3.5 Exterior Forms
5.4 Calculus on Manifolds
5.4.1 Differentiable Mappings of Manifolds
5.4.2 Integral Curves of Vector Fields
5.4.3 Exterior Product of One-Forms
5.4.4 The Exterior Derivative
5.4.5 Exterior Derivative and Vectors in R3
5.5 Hamilton-Jacobi and Lagrangian Mechanics
5.5.1 Coordinate Manifold Q, Velocity Space TQ, and Phase Space T*Q
5.5.2 The Canonical One-Form on Phase Space
5.5.3 The Canonical, Symplectic Two-Form on M
5.5.4 Symplectic Two-Form and Darboux''s Theorem
5.5.5 The Canonical Equations
5.5.6 The Poisson Bracket
5.5.7 Time-Dependent Hamiltonian Systems
5.6 Lagrangian Mechanics and Lagrange Equations
5.6.1 The Relation Between the Two Formulations of Mechanics
5.6.2 The Lagrangian Two-Form
5.6.3 Energy Function on TQ and Lagrangian Vector Field
5.6.4 Vector Fields on Velocity Space TQ and Lagrange Equations
5.6.5 The Legendre Transformation and the Correspondence of Lagrangian and Hamiltonian Functions
5.7 Riemannian Manifolds in Mechanics
5.7.1 Affine Connection and Parallel Transport
5.7.2 Parallel Vector Fields and Geodesics
5.7.3 Geodesics as Solutions of Euler-Lagrange Equations
5.7.4 Example: Force-Free Asymmetric Top
6. Stability and Chaos
6.1 Qualitative Dynamics
6.2 Vector Fields as Dynamical Systems
6.2.1 Some Definitions of Vector Fields and Their Integral Curves
6.2.2 Equilibrium Positions and Linearization of Vector Fields
6.2.3 Stability of Equilibrium Positions
6.2.4 Critical Points of Hamiltonian Vector Fields
6.2.5 Stability and Instability of the Free Top
6.3 Long-Term Behavior of Dynamical Flows and Dependence on External Parameters
6.3.1 Flows in Phase Space
6.3.2 More General Criteria for Stability
6.3.3 Attractors
6.3.4 The Poincar6 Mapping
6.3.5 Bifurcations of Flows at Critical Points
6.3.6 Bifurcations of Periodic Orbits
6.4 Deterministic Chaos
6.4.1 Iterative Mappings in One Dimension
6.4.2 Qualitative Definitions of Deterministic Chaos
6.4.3 An Example: The Logistic Equation
6.5 Quantitative Measures of Deterministic Chaos
6.5.1 Routes to Chaos
6.5.2 Liapunov Characteristic Exponents
6.5.3 Strange Attractors
6.6 Chaotic Motions in Celestial Mechanics
6.6.1 Rotational Dynamics of Planetary Satellites
6.6.2 Orbital Dynamics of Asteroids with Chaotic Behavior
7. Continuous Systems
7.1 Discrete and Continuous Systems
7.2 Transition to the Continuous System
7.3 Hamilton''s Variational Principle for Continuous Systems
7.4 Canonically Conjugate Momentum and Hamiltonian Density.
7.5 Example: The Pendulum Chain
7.6 Comments and Outlook
Exercises
Chapter 1: Elementary Newtonian Mechanics
Chapter 2: The Principles of Canonical Mechanics
Chapter 3: The Mechanics of Rigid Bodies
Chapter 4: Relativistic Mechanics
Chapter 5: Geometric Aspects of Mechanics
Chapter 6: Stability and Chaos
Solution of Exercises
Chapter I: Elementary Newtonian Mechanics
Chapter 2: The Principles of Canonical Mechanics
Chapter 3: The Mechanics of Rigid Bodies
Chapter 4: Relativistic Mechanics
Chapter 5: Geometric Aspects of Mechanics
Chapter 6: Stability and Chaos
Appendix
A. Some Mathematical Notions
B. Historical Notes
Bibliography
Index
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